Across science and engineering, the same template recurs: a designer is given a target behaviour and must find a physical object that realises it. A medicinal chemist is given a binding-affinity profile and must find a molecule whose density-functional-theory (DFT) energetics match it [1, 2, 3]; a photonics engineer is given a target far-field radiation pattern and must find a metasurface whose Maxwell-solver response matches it [4, 5]; a crystallographer is given a desired band gap and must find a periodic crystal whose first-principles spectrum matches it [6, 7, 8]. Inverse design asks: given a target property vector \( p^\star\) , find a design \( x^\star \in \mathcal{X}\) such that \( f(x^\star) \approx p^\star\) , where \( f\) is an expensive black-box oracle (e.g., a physical assay or simulator) and a feasibility constraint \( g(x) \le 0\) encodes stability requirements. Three structural obstacles make direct search on \( \mathcal{X}\) intractable. (i) \( \mathcal{X}\) is discrete and non-Euclidean (graphs of atoms, fields of crystallographic orientations), so gradient methods do not apply through a simulator that has no usable adjoint [9]. (ii) most points in \( \mathcal{X}\) are physically invalid, and feasibility cannot be expressed as a simple box, so unguided sampling almost never yields a candidate that oracle will run to convergence [10]. (iii) Each evaluation \( f(x)\) costs hours, with \( 10^2\) –\( 10^3\) evaluation budgets that rules out brute-force sampling, GAN-inversion-by-backpropagation [11, 12], and high-dimensional Bayesian optimization (BO) over the raw space [13, 14, 15]. Prior work side-steps the simulator by training a fast surrogate [16, 17, 6, 7], trading simulator cost for surrogate-fidelity risk that breaks whenever the surrogate is asked to extrapolate past its training distribution.

To ground this framework, we tackle an industrial and scientific challenge that falls entirely out of the ’cheap-oracle’ assumption dominating existing latent-optimization research: target-driven inverse design of polycrystalline Magnesium (Mg) alloy microstructures and textures. The protagonist is the engineer with a target stress–strain curve, eg. a structural designer specifying a Mg-alloy AZ31 component for an automotive crash member [18]; a clinician specifying a bioresorbable Mg–Gd bone screw whose yield strength must match cortical bone over a degradation window [19]; an aerospace designer chasing an anisotropic high-toughness texture in a structural panel [20]. In every case the deliverable is not a microstructure image but an arrangement of grains and crystallographic orientations that, when simulated through a crystal-plasticity (CP) oracle such as Damask [21], returns a property tuple \( (\sigma_y, n, K, \sigma_u)\) within tolerance of the target. Microstructure inverse design is a hard problem because the design space comprises of grains with symmetric-crystallographic orientaions [22, 23] and anisotropic, twin-dominated, and texture-coupled plasticity which causes visually similar microstructures to yield vastly different stress–strain responses [24]; and a single \( 300{\times}300\) CP simulation takes \( \sim\!6\) minutes and risks divergence [21]. Inverse microstructure and texture design are therefore recognised as ill-conditioned, high-dimensional, and data-starved [25, 26].

Why no off-the-shelf recipe applies? To bypass the intractability of raw search space, one could theoretically map the problem into a continuous latent search space with a pretrained generator \( \mathcal{D} : \mathbb{R}^d \to \mathcal{X}\) . In such a setup, decoded samples lie near a learned manifold, the latent prior yields a free box constraint \( \mathcal{Z} = [-B, B]^d\) , and the optimization objective \( J \circ f \circ \mathcal{D}\) is well-defined. This theoretically enables pure target-driven design, where optimal orientations and grain morphologies emerge directly from stress–strain requirements [27]. In practice, however, existing studies assume cheap oracles [28, 1, 29, 30], high-dimensional BO lacks generative priors [13, 14, 31], and recent crystal models [6, 7, 8] are built for sampling, not inverse optimization. None can reliably navigate an expensive, occasionally divergent simulator while enforcing valid material-grain orientation fields.

What is new. The latent representation is an enabling component, not our contribution; the methodological claim concerns the optimizer. Existing active-learning optimizers assume that every query returns a valid value and search generic, isotropic regions. Meridian couples two mechanisms not jointly present in any of the seven baselines: failure-aware feasibility learning, a classifier trained on failed decode/codec/solver evaluations that steers acquisition away from likely divergence, and manifold-aware local search, a surrogate-shaped active-subspace trust region with a latent-shell projection that keeps candidates where the decoder yields valid microstructures. To our knowledge, Co-PiLOT is also the first closed loop coupling an image-domain generative prior to a crystal-plasticity solver under a \( 160\) -simulation budget.

Contributions. We present Co-PiLOT (Constrained Physics-informed Latent Optimization for Target-driven inverse design), a framework that lifts inverse design into a pretrained generative latent space and runs a constrained physics-informed black-box optimizer there.

1. General framework (Sec. 3). We formulate inverse design as a 5-tuple \( (\mathcal{D}, f, J, \mathcal{C}, \mathcal{O})\) with independently swappable decoder, oracle, objective, constraint set, and optimizer. The abstraction is domain-agnostic; we demonstrate it on one domain (Mg-alloy microstructures) and leave other instantiations to future work.
2. Microstructure encoder–decoder family (Sec. 3.1). A shared Vision Transformer (ViT) based encoder is paired with a primary Flow Matching Diffusion Transfomer (FMDiT) based decoder, trained jointly on \( 81{,}756\) EBSD-derived microstructures across 17 Mg-alloy classes.
3. Orientation codec (Sec. 3.2). An invertible map between RGB images and orientation fields with a closed-form decode and a grain-segmentation Dream.3D writer that is used by the CP solver. It achieves \( {\sim}0.7^\circ\) round-trip misorientation, below the \( 5^\circ\) grain-boundary threshold [32, 33].
4. Meridian, a latent optimizer (Sec. 3.4). We developed a combination of a deep-kernel Gaussian Process surrogate, a shared-trunk feasibility classifier, an active-subspace trust region with latent-shell projection, and a DPP batch selector; against seven baselines under a matched budget, ViT-FMDiT-\( 768\) with Meridian reaches the best target-driven and toughness-weighted objectives.

The 5-tuple abstraction is domain-agnostic: in principle any setting with a pretrained decoder for \( \mathcal{X}\) and a slow oracle scoring \( \mathcal{D}(z)\) is in scope (molecules with a VAE and a DFT oracle [1], photonic devices with an EM solver [34], topology optimization with an FEM oracle [35]), but we demonstrate it only on Mg-alloy microstructures. All code is released (Sec. 6).

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High-dimensional latent optimization. Latent-space optimization was popularized by [1], who showed that a learned continuous representation can make structured design amenable to gradient-based and Bayesian optimization. Follow-up latent-BO methods add weighted retraining, trust regions, and constrained or multi-objective variants [28, 29, 30, 36], but they are largely evaluated with cheap neural property predictors rather than expensive physical simulators. High-dimensional BO methods such as TuRBO, BAxUS, and SaasBO improve sample efficiency in large ambient spaces,[13, 14, 31] yet they do not exploit generative priors or explicitly address simulator divergence and structured feasibility, which are central in our setting. Classical black-box optimizers such as CMA-ES remain strong generic baselines [37], although full covariance adaptation scales poorly in high dimensions; we include it in Sec. 5.

Generative inverse design of materials. Generative models have become a standard tool for inverse design in materials and related physical systems [10]. Early materials work used GANs to generate microstructures and couple them to regressors or Bayesian optimization [16, 17], while more recent studies perform inverse design of dual-phase steel microstructures with generative models and BO [38], inverse design of spinodoid architected materials in the small-data regime via BO [39], or use diffusion models for nonlinear mechanical metamaterials and multi-material structures [40, 41]. For polycrystals, even representing microstructure for design remains open [42], and inverse microstructure and texture design rely on informative low-dimensional priors or conditional normalizing flows in active-learning loops [25, 26]. In crystals, CDVAE, DiffCSP, and MatterGen show that modern generative models can capture complex structure distributions [6, 8, 7], but these methods are primarily built for sampling or one-shot conditional generation rather than iterative optimization under a strict expensive-oracle budget.

Black-box optimization around generators. Our work is closest in spirit to methods that place an outer optimization loop around a pretrained generator, as in latent BO for molecules [1, 28], photonic inverse design with neural generators and EM solvers [4, 5, 34], and GAN-based topology optimization with FEM [35, 40]. Diffusion black-box optimizers (DDOM [43]) and reward fine-tuning of diffusion models (DDPO [44], SEIKO [45]) assume offline data or far larger query budgets; optimizing over learned models can exploit adversarial modes [46], and coupling generative models to PDE physics remains frontier work [47]. The key gap is that these settings typically rely on cheaper solvers, smoother objectives, or direct conditional generation. Co-PiLOT instead targets the harsher regime of constrained latent optimization with a slow, occasionally divergent physics solver, and therefore emphasizes calibrated surrogates, feasibility modeling, and robust search in pretrained latent spaces.

The central act of Co-PiLOT is the tight coupling of a pretrained microstructure decoder \( \mathcal{D}\) (Sec. 3.1) with a physics-informed optimizer \( \mathcal{O}\) (Sec. 3.3) via an orientation codec ensuring crystallographic validity (Sec. 3.2). Operating hand-in-hand inside a single loop (Fig. 1), the decoder turns a latent vector \( z{\in}\mathcal{Z}\) into a candidate design, while the optimizer proposes the next \( z\) to query based on a scalar fitness \( J\) returned by an oracle simulator \( f\) (Sec. 3.4).

Problem formulation. Let \( \mathcal{X}\) denote a structured design space (microstructures, molecules, \dots), \( f : \mathcal{X} \to \mathbb{R}^k\) an expensive oracle returning a property tuple \( p = f(x) \in \mathbb{R}^k\) , \( p^\star \in \mathbb{R}^k\) a target vector, \( J : \mathbb{R}^k \to \mathbb{R}\) a scalar objective combining target distance and physics priors, and \( g : \mathcal{X} \to \mathbb{R}^m\) feasibility constraints. Given a pretrained encoder–decoder pair \( (\mathcal{E}, \mathcal{D})\) with \( \mathcal{D} : \mathbb{R}^d \to \mathcal{X}\) , Co-PiLOT maps \( \mathcal{X}\) into the latent box \( \mathcal{Z} = [-B, B]^d\) inherited from the latent prior and solves

This reformulation has three structural advantages. (i) Deliberate validity as \( \mathcal{D}(z)\) is always a plausible design. (ii) Box constraint gives a natural prior \( \mathcal{Z}\) . (iii) Regularised search space as the decoder restricts the optimizer to a learned manifold, resulting in structured modeling target than direct search over the raw \( \mathcal{X}\) . The materials instantiation (Fig. 1b) dictates the anatomy of this section.

Where physics enters Co-PiLOT. “Physics-informed” follows the physics–ML taxonomies of [48, 68], not PINN-style residual losses. Physics enters at two levels (Fig. 1b). (1) Physics-informed objective: \( J\) is computed from a crystal-plasticity simulation at every iteration (\( J_{V2}\) measures distance to the target properties, \( J_{V1}\) adds a ductility floor, both penalise unphysical grain counts), and Meridian’s feasibility model learns simulator divergence (Secs. 3.3, 3.4); this is knowledge integration via the learning objective [48]. (2) Physics-augmented learned components: the encoder–decoder is trained on codec-encoded EBSD maps whose every pixel is a valid HCP orientation, so the latent manifold is a manifold of physically realisable microstructures, and Meridian’s surrogate is warm-started on \( 100\) crystal-plasticity simulations rather than a cold random prior (Secs. 3.2, 5).

3.1 Implementation of Encoder–Decoder (\( \mathcal{D}\) )

Co-PiLOT is built around a continually pre-trained stack: a ViT-H/14 encoder [49, 50] and diffusion decoders- SDXL [51], SD3.5 [52], and DiT-XL/2 [53], that are pre-trained on massive natural-image datasets. We posit this prior is very effective for microstructure reconstruction on a very thin bottlneck \( z\) . Even a \( d{=}512\) bottleneck (\( {\sim}1500{\times}\) compression) recovers grain morphology, crystallographic colour, and property statistics that are compareable to a from-scratch decoder (Sec. 4).

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A single shared encoder architecture. A ViT-H/14 trunk maps \( 512^2{\times}3\) orientation-as-RGB microstructure images to \( 1369\) patch tokens plus [CLS] token. 2B weights are reused with the lower \( 16\) blocks frozen and the upper \( 16\) unfrozen (Fig. 2(a)), retaining the natural-image prior while letting the top of the trunk specialise to EBSD imagery. The [CLS]-only readout is replaced by a four-query Attention Pooler (cross-attention over all \( 1370\) tokens, query self-attention, per-query FFN, then merged) followed by a two-layer MLP that maps to bottleneck \( z\in\mathbb{R}^{d}\) , with \( d{=}512\) as the headline; \( d\in{768,1024}\) are reported as the bottleneck-capacity ablation in Sec. 4.

Three diffusion decoders are each conditioned solely on the bottleneck, \( z\) . The pre-trained backbones are frozen and lightweight adaptors are introduced that translate \( z\) into the native conditioning format of each model (Fig. 2b). FM-DiT (frozen SD3.5 MMDiT [52]): We introduce a LatentToTokens adaptor using \( 16\) queries and \( 5\) QK-normed AdaLN-Zero blocks [53]. This generates \( 16{\times}4096\) tokens for the joint attention layers, while a Pooled Projection head provides the \( 2048\) -D global embedding. SDXL (frozen denoising UNet [51]): The latent \( z\) is mapped to \( 77{\times}2048\) tokens via \( 3\) AdaLN-Zero blocks. We initialize the outer layers with \( 0.10\) Xavier gain, as standard zero-initialization collapses the conditioning heads. A Pooled Projection head maps \( z\) to the \( 1280\) -D embedding, and we apply classifier-free guidance [54]. DiT (DiT-XL/2 [53]): Unlike the others, this model is fully trainable, as freezing the backbone fails during the \( 256{\to}512\) resolution transfer. We repurpose the DiT class-embedding slot to inject \( z\) using a ClassProj head. Additionally, \( 16{\times}1152\) spatial tokens are injected via zero-initialized residuals at blocks \( {0,7,14,21}\) . Architectures for SDXL and DiT are deferred to App. Fig. 8. We also include a pixel-space VQGAN [55] with a separate codebook as a non-\( z\) -bottleneck baseline.

Joint training. Encoder, conditioning adaptors, and backbone (for ViT-DiT) are trained jointly under a composite of latent-space generative terms, a latent-alignment regulariser, and a pixel-space frequency term that is gated to small denoising times \( t\) . The loss function per batch, with \( \hat x_0\) as the clean-latent estimate, \( z\) as the bottleneck, and \( z'\) as its augmentation-twin embedding is given as,

\[ \begin{equation} \mathcal{L} \;=\; \lambda_{\mathrm{diff}}\,\mathcal{L}_{\mathrm{diff}} + \lambda_{\mathrm{rec}}\,\mathcal{L}_{\mathrm{rec}} + \lambda_{\mathrm{con}}\,\mathcal{L}_{\mathrm{con}} + \lambda_{\mathrm{vic}}\,\mathcal{L}_{\mathrm{vic}} + \lambda_{\mathrm{freq}}\,\mathcal{L}_{\mathrm{freq}}, \end{equation} \]

(2)

where \( \mathcal{L}_{\mathrm{diff}}\) is the backbone’s native objective — the rectified-flow velocity loss \( \lVert v_\theta(x_t,t,z) - (\epsilon - x_0)\rVert_2^2\) for ViT-FMDiT [56] and the standard MSE loss for ViT-SDXL and ViT-DiT; \( \mathcal{L}_{\mathrm{rec}} = \mathbb{E}_t\bigl[(1{-}t)^2\,\lVert\hat x_0 - x_0\rVert_2^2\bigr]\) is a timestep-weighted clean-latent reconstruction term that emphasises low-noise steps; \( \mathcal{L}_{\mathrm{con}}\) is an InfoNCE-style contrastive loss over \( {z, z'}\) pairs; \( \mathcal{L}_{\mathrm{vic}}\) is a VICReg variance–covariance regulariser [57] that keeps the bottleneck non-collapsed; \( \mathcal{L}_{\mathrm{freq}} = \lVert \log(1{+}|\mathcal{F}(\hat I)|) - \log(1{+}|\mathcal{F}(I)|)\rVert_1\) is an FFT-amplitude pixel loss with a radial high-frequency mask, evaluated on a small subsample of the batch. The five \( \lambda\) s follow one fixed LossWeightScheduler schedule, reused for every ViT-FMDiT model and never tuned per experiment: \( \lambda_{\mathrm{diff}}\) rises \( 0.5{\to}1.0\) while \( \lambda_{\mathrm{rec}}\) and \( \lambda_{\mathrm{con}}\) decay, and the two regularisers enter with a delayed onset (App. C); perturbing it moves held-out PSNR by at most \( 0.09\)  dB (Tab. 6). A two-phase schedule freezes the encoder for the first seven epochs and unfreezes its last \( 16\) blocks at epoch \( 8\) , in step units to absorb the topology change when the optimizer is rebuilt and the cosine schedule recomputed. More details on the implementations can be found in App. C.

3.2 Orientation codec (\( \Psi\) )

3.3 Crystal plasticity simulation pipeline and objectives (\( f\circ\mathcal{D}\) , \( J\) )

3.4 Optimizer: Meridian (Ours) (\( \mathcal{O}\) )

The encoder–decoder models described in Sec. 3.1 are trained once and then frozen for all downstream optimization experiments in Sec. 5. Expanding on the data corpus of [24], we assemble \( 81{,}756\) training and \( 9{,}084\) held-out test RGB microstructures; this expansion was achieved through geometry and physics-aware oversampling, the details of which are outside the scope of this paper and will be documented in a future publication. All the models from Tab. 1 are trained on \( 4\times\) H200 GPUs, and a standard \( 30\) -epoch run on the training split takes roughly \( 12\) hours. The VQGAN reference, FM-DiT at \( z\in{512,768,1024}\) , and SDXL all use epoch-\( 30\) checkpoints; DiT is substantially heavier because its denoiser backbone is unfrozen, carrying about \( 1.17\) B trainable parameters versus roughly \( 381\) M–\( 391\) M for the FMDiT family, and the comparison therefore uses the latest available epoch-\( 20\) checkpoint from the same training regime. We score reconstruction quality with three computer-vision metrics and two material science metrics. FID is the distribution-level realism score, MS-SSIM captures multi-scale structural similarity, and LPIPS captures perceptual distance [62, 63, 64]. For materials fidelity, we report \( G_{\mathrm{m}}\) for grain recovery (fraction of original grains recovered at IoU \( \ge 0.3\) , capturing grain size and aspect-ratio agreement) and \( \Delta_{\mathrm{ori}}\) for crystallography (mean disorientation between IoU-matched per grains per-class mean quaternions).

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ViT-DiT achieves the best FID, while the ViT-FMDiT family (specifically at \( d{=}768\) ) achieves the best \( G_{\mathrm{m}}\) —indicating the strongest grain-level recovery in size and shape—alongside the best MS-SSIM, LPIPS, and \( \Delta_{\mathrm{ori}}\) among bottlenecked decoders (Tab. 1). ViT-FMDiT’s performance is stable across bottleneck size, allowing the optimization loop (Sec. 5) to use a tighter bottleneck without sacrificing fidelity. Conversely, ViT-SDXL struggles to adapt its text-conditioned UNet to a 512-D continuous bottleneck, yielding the worst metrics (FID \( \sim\!108\) ) and visibly washed-out grains in Fig. 3. An unbottlenecked ViT-VQGAN provides an upper bound for reconstruction, while the orientation codec establishes the error floor. The codec’s \( \Delta_{\mathrm{ori}}\) error is safely within CP tolerances. While the bottlenecked decoders preserve macroscopic morphology and class statistics (e.g., AZ31, Mg-10Gd), their orientation space is non-isometric and symmetry-blind. This introduces a \( {\sim}53^\circ\) per-pixel disorientation into the end-to-end round-trip. This residual error acts as a class-preserving reshuffle rather than a physical violation as the decoded grains remain in the HCP fundamental zone, and their texture statistics (quaternion mean, pole-figure spread) matches the target class. Therefore, we carry ViT{-}DiT and ViT{-}FMDiT with \( d\in{512,768,1024}\) into the Co-PiLOT loop as robust generators of statistically valid microstructures rather than exact reconstructors.

E2 tests the central claim : Can Co-PiLOT find statistically valid AZ31 microstructures whose simulated stress–strain summaries match a user-specified mechanical target? We use the optimizer suite from Sec. 3.4 and the trained decoders from Sec. 4. The target, \( p^\star=(\sigma_y^\star,n^\star,K^\star,\sigma_u^\star)=(160\,\mathrm{MPa},0.302,604\,\mathrm{MPa},316\,\mathrm{MPa})\) is set. Each cell uses \( 5\) independent runs and the same \( 100\) (\( z_\mathbf{init}\) <-> Damask evaluation) seed cache for warm starting the surrogate model. The optimizer evaluates \( 40\) iterations with batch size \( 4\) , i.e. \( 160\) new simulator evaluations per run. Average runtime per evaluation on a \( 700\) W H200 is \( 8.46\) hours with a GPU energy estimate of approx \( 5.93\) kWh across the full sweep. Table 2 identifies ViT{-}FMDiT-\( 768\) with Meridian as the strongest target-driven configuration. After 144 evaluations (Fig. 4), it yields the closest overall target match (\( J_{V2}=-0.132\pm0.004\) , \( \sigma_y=159.1\) MPa, \( n=0.299\) , \( \sigma_u=314.0\) MPa), successfully falling within the AZ31 material class. This success highlights a key principle: the optimizer and decoder must align. The \( 768\) -D bottleneck offers the ideal trade-off—more expressive than \( 512\) -D, yet dense enough for local surrogate modeling. Conversely, the \( 1024\) -D bottleneck adds expressivity but becomes too sparse under a \( 160\) -evaluation budget, so larger latents do not automatically improve inverse design. This reinforces our core conclusion: successful inverse design requires an optimizer whose inductive biases match the encoder-decoder’s latent space (see App. A for \( J_{V1}\) ).

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Table 2. Optimization Results: Target-Driven, \( J_{V2}\) (\( n_{\mathrm{seed}}= 5\) ).
Green = ours. \( ^*\) = best per block. \( ^\dagger\) = best overall. # sims = evals to best. Below line: added baselines.

Decoder	Optimizer	\( J_{V2}\uparrow\)	\( \sigma_y\)	\( n\)	\( \sigma_u\)	# sims
ViT-DiT	Meridian	\( -0.156 \pm 0.005^*\)	\( 155.9\)	\( 0.305\)	\( 313.9\)	\( 160\)
	Dante	\( -0.163 \pm 0.009\)	\( 155.8\)	\( 0.305\)	\( 312.9\)	\( 104\)
	TuRBO	\( -0.168 \pm 0.007\)	\( 154.1\)	\( 0.301\)	\( 307.7\)	\( 152\)
	BAxUS	\( -0.168 \pm 0.009\)	\( 154.8\)	\( 0.303\)	\( 310.6\)	\( 152\)
ViT-FMDiT	Meridian	\( -0.152 \pm 0.004^*\)	\( 156.1\)	\( 0.302\)	\( 312.1\)	\( 96\)
	Dante	\( -0.161 \pm 0.016\)	\( 157.1\)	\( 0.303\)	\( 314.4\)	\( 156\)
	TuRBO	\( -0.167 \pm 0.009\)	\( 154.4\)	\( 0.301\)	\( 308.4\)	\( 160\)
	BAxUS	\( -0.162 \pm 0.005\)	\( 155.1\)	\( 0.303\)	\( 311.1\)	\( 144\)
ViT-FMDiT 768	Meridian\( ^\dagger\)	\( \mathbf{-0.132 \pm 0.004}^*\)	\( 159.1\)	\( 0.299\)	\( 314.0\)	\( 144\)
	Dante	\( -0.140 \pm 0.004\)	\( 157.7\)	\( 0.301\)	\( 313.1\)	\( 140\)
	TuRBO	\( -0.151 \pm 0.009\)	\( 158.5\)	\( 0.306\)	\( 318.5\)	\( 148\)
	BAxUS	\( -0.146 \pm 0.004\)	\( 157.7\)	\( 0.305\)	\( 316.5\)	\( 156\)
	CMA-ES	\( -0.144 \pm 0.002\)	\( 156.6\)	\( 0.302\)	\( 313.1\)	\( 147\)
	DDOM	\( -0.136 \pm 0.008\)	\( 157.9\)	\( 0.301\)	\( 313.1\)	\( 123\)
	SEIKO	\( -0.165 \pm 0.009\)	\( 153.5\)	\( 0.304\)	\( 309.5\)	\( 133\)
	DDPO	\( -0.170 \pm 0.003\)	\( 152.9\)	\( 0.305\)	\( 309.7\)	\( 144\)
ViT-FMDiT 1024	Meridian	\( -0.150 \pm 0.005^*\)	\( 157.0\)	\( 0.304\)	\( 314.0\)	\( 152\)
	Dante	\( -0.168 \pm 0.004\)	\( 153.6\)	\( 0.304\)	\( 308.8\)	\( 124\)
	TuRBO	\( -0.156 \pm 0.009\)	\( 157.1\)	\( 0.305\)	\( 315.0\)	\( 140\)
	BAxUS	\( -0.164 \pm 0.008\)	\( 155.3\)	\( 0.304\)	\( 311.6\)	\( 124\)
	CMA-ES	\( -0.167 \pm 0.005\)	\( 154.0\)	\( 0.307\)	\( 312.6\)	\( 144\)
	DDOM	\( -0.154 \pm 0.009\)	\( 155.4\)	\( 0.305\)	\( 313.0\)	\( 111\)
	SEIKO	\( -0.174 \pm 0.011\)	\( 152.6\)	\( 0.307\)	\( 310.4\)	\( 111\)
	DDPO	\( -0.176 \pm 0.004\)	\( 152.7\)	\( 0.309\)	\( 312.2\)	\( 155\)

Extended baselines. On ViT-FMDiT-\( 768\) /\( 1024\) we also ran CMA-ES, DDOM, SEIKO, and DDPO under the same protocol (Tab. 2, lower rows). Meridian stays best (\( -0.132\) /\( -0.150\) ); DDOM (\( -0.136\) /\( -0.154\) ) is within one standard deviation but never ahead, as it models neither feasibility nor trust-region geometry, and surrogate-free CMA-ES trails (\( -0.144\) /\( -0.167\) ) because it must spend simulator calls to learn the landscape. SEIKO and DDPO, which fine-tune the decoder toward average reward, do not beat the warm-start incumbent within \( 160\) evaluations, \( 160\) –\( 360\times\) fewer queries than DDPO’s reference configurations. The middle of the ranking reorders between decoders while the top two do not, so the result reflects the method rather than one bottleneck width. Across four decoders and two objectives, Meridian is best in all eight studies; batch-rule and sensitivity ablations are in App. B.

Limitations. Co-PiLOT is limited by its decoder as it can only search microstructures that the learned prior can express. The decoder is also a class-conditional generator of statistically valid AZ31 microstructures, not a faithful per-pixel orientation reconstructor; although the codec is sub-degree accurate, the encoder–decoder is still non-isometric in crystallographic rotation distance. The Damask oracle is HCP phenopower-law plasticity, no damage, isothermal loading, and the loading paths studied here, therefore, inverse-designed microstructures require multi-loading and experimental validation. This paper does not show transfer to other polycrystalline materials, property sets, or physical domains, and compares optimizers at a single \( 160\) -evaluation budget; harder inverse-design problems are future work. We are addressing these issues by expanding validation and training \( \mathcal{E}\) –\( \mathcal{D}\) with orientation-specific losses such as HCP-symmetry-aware grain-level quaternion losses and texture-statistics penalties.

Highlights and broader impact. Co-PiLOT turns expensive simulator-driven inverse design into active search over a learned generative design space; to our knowledge, no similar end-to-end pipeline previously existed for this regime. The decoders are evaluated by FID, MS-SSIM, LPIPS, grain recovery, and disorientation against ground truth (Tab. 1) and Meridian against seven optimizers under a matched budget plus batch-rule and control-parameter ablations (Tab. 2, App. B). The codec makes image-domain generators usable inside a crystal-plasticity loop, and Meridian combines calibrated uncertainty, feasibility modeling, manifold-aware proposals, and diverse batches to reduce wasted simulator calls. On AZ31 design, ViT-FMDiT-\( 768\) with Meridian gives the strongest target-driven result and remains effective on the toughness objective. The abstraction may extend to other physical systems with learned priors and slow oracles, which remains to be shown. Code: https://github.com/mahishguru/microstructure-encoder-decoder (encoder–decoder), https://github.com/mahishguru/orientation-codec (orientation codec), and https://github.com/mahishguru/meridian (Meridian, Damask harness).

Acknowledgements We thank the anonymous reviewers and the area chair for comments that improved the evaluation of this work.

Funding. We acknowledge Helmholtz-Zentrum Hereon for providing computational infrastructure and institutional support. Additional support was provided by the Joachim Herz Stiftung through an Add-on Fellowship.

Competing interests. The authors declare no competing interests.

This appendix reports the toughness-weighted objective results that complement the target-driven E2 results in Sec. 5. The protocol is unchanged: the shared seed cache is used only for warm start, all curves and tables exclude that cache, and each run receives \( 160\) optimizer-phase simulator evaluations. Average wall-clock runtime for these \( J_{V1}\) runs was \( 8.48\) hours on an H200 machine, corresponding to an estimated \( 5.94\) kWh of GPU energy per run under the same \( 700\) W TDP accounting used in the main text.

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Table 3. Optimization Results: Toughness-Weighted, \( J_{V1}\) (\( n_{\mathrm{seed}}= 5\) ).

Decoder	Optimizer	\( J_{V1}\uparrow\)	\( \sigma_y\)	\( n\)	\( \sigma_u\)	# sims
ViT-DiT	Meridian	\( 0.930 \pm 0.003^*\)	\( 154.3\)	\( 0.308\)	\( 314.0\)	\( 144\)
	Dante	\( 0.929 \pm 0.010\)	\( 155.6\)	\( 0.305\)	\( 313.4\)	\( 148\)
	TuRBO	\( 0.915 \pm 0.007\)	\( 153.8\)	\( 0.306\)	\( 311.3\)	\( 112\)
	BAxUS	\( 0.918 \pm 0.002\)	\( 151.8\)	\( 0.312\)	\( 314.2\)	\( 160\)
ViT-FMDiT	Meridian	\( 0.943 \pm 0.011^*\)	\( 156.8\)	\( 0.306\)	\( 316.1\)	\( 124\)
	Dante	\( 0.923 \pm 0.008\)	\( 155.2\)	\( 0.305\)	\( 313.0\)	\( 152\)
	TuRBO	\( 0.931 \pm 0.003\)	\( 154.0\)	\( 0.309\)	\( 314.2\)	\( 100\)
	BAxUS	\( 0.933 \pm 0.006\)	\( 156.3\)	\( 0.301\)	\( 311.7\)	\( 100\)
ViT-FMDiT 768	Meridian\( ^\dagger\)	\( \mathbf{0.961 \pm 0.008}^*\)	\( 157.9\)	\( 0.305\)	\( 316.8\)	\( 112\)
	Dante	\( 0.954 \pm 0.014\)	\( 159.8\)	\( 0.300\)	\( 316.6\)	\( 140\)
	TuRBO	\( 0.948 \pm 0.007\)	\( 156.1\)	\( 0.309\)	\( 317.9\)	\( 160\)
	BAxUS	\( 0.953 \pm 0.004\)	\( 157.2\)	\( 0.304\)	\( 315.1\)	\( 116\)
ViT-FMDiT 1024	Meridian	\( 0.952 \pm 0.005^*\)	\( 156.2\)	\( 0.308\)	\( 317.6\)	\( 160\)
	Dante	\( 0.923 \pm 0.006\)	\( 153.5\)	\( 0.309\)	\( 314.0\)	\( 156\)
	TuRBO	\( 0.944 \pm 0.010\)	\( 157.2\)	\( 0.305\)	\( 315.3\)	\( 140\)
	BAxUS	\( 0.936 \pm 0.011\)	\( 155.8\)	\( 0.308\)	\( 316.7\)	\( 128\)
Green = our method (Meridian). \( ^*\) = best per decoder block. \( ^\dagger\) = best overall.

The objective changes the design question from hitting a prescribed stress–strain tuple to finding high-strength, high-plastic-work microstructures without dropping below the hardening floor. Figure 6 and Table 3 show that the same decoder–optimizer pairing remains the most effective overall: ViT{-}FMDiT-\( 768\) with Meridian reaches \( J_{V1}=0.961\pm0.008\) , with \( \sigma_y=157.9\) MPa, \( n=0.305\) , \( \sigma_u=316.8\) MPa, and \( 112\) optimizer-phase simulations to the best value. Compared with the target-driven case, the best solution now moves slightly toward higher ultimate strength while preserving hardening, as expected for a toughness-weighted score.

The \( 1024\) -D appendix rows mirror the main-text interpretation. This does not indicate a defective decoder; rather, the wider bottleneck exposes isolated texture pockets that are difficult to model globally from \( 160\) calls. Sparse local perturbations and low-dimensional PCA-aligned subspace moves can exploit such pockets without fitting the whole ambient latent at once, which explains why the best \( J_{V1}\) rows for ViT{-}FMDiT-\( 1024\) come from those search biases. Across both objectives, the consistent conclusion is that ViT{-}FMDiT-\( 768\) gives the best operating point for Co-PiLOT: enough latent capacity to express useful AZ31 textures, but not so much that the expensive-oracle optimizer loses sample efficiency.

Ranking consistency. Our conclusions are ranking statements (which optimizer and which decoder are best), so we test whether the rankings survive variations in the control parameters. Across four decoders and two objectives (Tabs. 2, 3), Meridian attains the best mean objective in all eight studies; under a null of uniformly random ranking among the four optimizers run on every decoder, this occurs with probability \( (1/4)^8 \approx 1.5\times10^{-5}\) . Symmetrically, ViT-FMDiT-\( 768\) is the best decoder in all eight optimizer–objective combinations. The ViT-DiT decoder was trained under a materially different loss-weight configuration from the ViT-FMDiT decoders (\( \lambda_{\mathrm{vic}}\) \( 0.01\) vs. \( 0.05\) , \( \lambda_{\mathrm{freq}}\) \( 0.1\) vs. \( 0.3\) , \( \lambda_{\mathrm{con}}\) starting at \( 0.0\) vs. \( 0.3\) , different onset/ramp epochs), so the optimizer ranking is not an artifact of one loss-weight specification.

DPP versus greedy top-\( k\) batch selection. We swap only Meridian’s DPP batch selector for plain top-\( k\) by acquisition score, holding every other component and the seed cache fixed (\( J_{V2}\) , \( 160\) evaluations, matched seeds; Tab. 4). These are separate matched runs, so values differ slightly from Tab. 2. On ViT-FMDiT-\( 768\) both rules reach the same mean quality (\( J_{V2}=-0.138\) ), so DPP does not raise the ceiling, but it reaches the plateau about \( 11\%\) of the budget earlier (\( 143\) vs. \( 160\) evaluations) with lower run-to-run spread (\( \pm0.007\) vs. \( \pm0.010\) ), and it eliminates a batch-collapse failure mode in which one top-\( k\) seed selected four near-duplicate candidates and made no progress after the first iteration. On ViT-FMDiT-\( 1024\) the two rules tie statistically (\( -0.165\pm0.010\) vs. \( -0.162\pm0.006\) , no collapsed seeds). Under an expensive oracle and a thin-shell latent manifold, batch diversity therefore buys sample efficiency and robustness rather than peak performance.

Sensitivity to Meridian’s control parameters. Meridian has its own settings: trust-region size and shrink rate, the latent-norm shell bound \( k_\sigma\) , and the batch-diversity term. We grouped repeated ViT-FMDiT-\( 768\) Meridian runs under one identical pair of objectives by the configuration they were produced under (Tab. 5). The evaluation budget is essentially unaffected: under \( J_{V2}\) every setting reaches its plateau at \( 104\) –\( 106\) of the \( 160\) evaluations, and no setting stalls, diverges, or fails to converge. The achieved optimum moves by about one seed-level standard deviation (\( 0.011\) on \( J_{V2}\) , against a per-setting seed spread of \( 0.005\) –\( 0.013\) ). The published setting is best on both objectives although it was tuned only on \( J_{V1}\) . The one setting with a systematic effect is the batch-diversity term, a claimed component of the method ablated above.

This appendix records the implementation detail behind the encoder–decoder family that the main text only sketches. The goal is not to restate Sec. 3.1, but to make the exact architectural, training, and validation choices reproducible: the shared encoder, the three conditioning interfaces, the composite training objective, and the bottleneck-capacity ablation.

Shared encoder — exact specification. The encoder is OpenCLIP ViT-H/14 (laion2b_s32b_b79k weights) [49, 50]: \( 32\) transformer blocks, hidden width \( 1280\) , patch size \( 14\) , and input resolution \( 512{\times}512\) , which yields \( 37{\times}37{=}1369\) patch tokens after bicubic interpolation of the original \( 16{\times}16\) positional embedding. We keep CLIP normalisation throughout rather than switching to ImageNet statistics. The freezing pattern is deliberately asymmetric: conv1, cls_token, the positional embedding, ln_pre, and the lower \( 16\) transformer blocks remain frozen, while only the upper \( 16\) blocks are trainable, with gradient checkpointing enabled there. This keeps most of the LAION prior intact while allowing the top of the trunk to adapt to EBSD imagery. The CLS readout is replaced by a multi-query attention pooler with \( 4\) learned queries (initialised \( \mathcal{N}(0,0.02)\) ), \( 8\) attention heads, query self-attention, and a per-query FFN, followed by a merge head \( \mathrm{Linear}(4D \to D)\) . The bottleneck MLP is \( \mathrm{Linear}(1280,1280)\to\mathrm{GELU}\to\mathrm{Linear}(1280,d)\to\mathrm{LayerNorm}\) with \( d \in {512,768,1024}\) .

Conditioning adaptors — exact specifications. All three decoders reuse the same bottleneck but expose different native conditioning interfaces, so each adaptor is designed to match the backbone it plugs into rather than to force a uniform abstraction. FM-DiT. LatentToTokens\( (d \to 16{\times}4096)\) uses \( 16\) learned queries, \( 5\) AdaLN-Zero blocks, and QK-normalised cross-attention against the pooled latent. Its outer \( \mathrm{proj\_out}\) is zero-initialised, which is stable here because flow matching learns a residual velocity field. The pooled head is \( \mathrm{Linear}(d,2048)\to\mathrm{SiLU}\to\mathrm{Linear}(2048,2048)\) , again with a zero-initialised last layer. The adaptor carries roughly \( 11\) M trainable parameters and the pooled head another \( 4\) M. The frozen backbone is stabilityai/stable-diffusion-3.5-medium; its VAE uses \( 16\) channels, \( 8{\times}\) downsampling, and scale factor \( 1.5305\) . Timesteps are sampled as \( t = \sigma(\mathcal{N}(0,1)+\log\,\mathrm{shift})\) with \( \mathrm{shift}{=}3.0\) , matching the published SD3.5 logit-normal schedule, and inference uses FlowMatchEulerDiscrete. SDXL. LatentToTokens\( (d \to 77{\times}2048)\) mirrors the original SDXL text sequence length with \( 77\) learned queries passed through \( 3\) AdaLN-Zero blocks. Here zero-initialisation was empirically unstable, so both the outer projection and the pooled head are initialised with Xavier gain \( 0.10\) ; runs with zero init produced dead conditioning heads. The pooled branch is \( \mathrm{Linear}(d,1280)\to\mathrm{SiLU}\to\mathrm{Linear}(1280,1280)\) feeding add_embedding, and a learned NullToken implements classifier-free guidance dropout with \( p_{\mathrm{drop}}{=}0.1\) . The trainable interface is roughly \( 8\) M parameters in the token adaptor, \( 0.8\) M in the pooled head, plus the \( 512\) -parameter null token. The SDXL UNet backbone remains frozen in bf16; training uses DDPM and inference EulerDiscrete. DiT. DiT-XL/2 (hidden size \( 1152\) , \( 28\) blocks, \( 16\) heads, patch size \( 2\) , VAE sd-vae-ft-mse) is fully trainable because freezing failed under the \( 256{\to}512\) transfer. The class-conditioning slot is repurposed through \( \mathrm{ClassProj}(d \to 1152) = \mathrm{Linear}\to\mathrm{GELU}\to\mathrm{Linear}\) with Xavier initialisation. A pair of forward pre-hooks (_CLIPInjector/_ZeroEmbedder) removes the original repeated class-embedding path and replaces it with a single top-of-stack injection of \( \mathrm{ClassProj}(z)\) . In parallel, LatentToSpatialTokens\( (d \to 16{\times}1152)\) emits \( 16\) spatial tokens that a SpatialCrossFuser inserts, with zero-initialised output projection, at blocks \( {0,7,14,21}\) . Training uses DDPM with \( 1000\) steps and linearly spaced \( \beta\) from \( 10^{-4}\) to \( 0.02\) ; inference uses \( 250\) -step DDIM.

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Composite loss and training schedule. Training combines the five terms introduced in Eq. (2) rather than relying on the backbone loss alone. The diffusion term \( \mathcal{L}_{\mathrm{diff}}\) is the rectified-flow velocity MSE \( \lVert v_\theta(x_t,t,z) - (\epsilon - x_0)\rVert_2^2\) for ViT-FMDiT and the standard \( \epsilon\) -prediction MSE \( \lVert\epsilon_\theta(x_t,t,z) - \epsilon\rVert_2^2\) for SDXL and ViT-DiT, always evaluated in the native VAE latent of the corresponding backbone. The reconstruction term is \( \mathcal{L}_{\mathrm{rec}} = \mathbb{E}_t[(1{-}t)^2\lVert\hat x_0-x_0\rVert_2^2]\) , which biases training toward cleaner timesteps; for the velocity parameterisation, \( \hat x_0 = x_t - t v_\theta\) . On top of this, we add a contrastive InfoNCE-style term \( \mathcal{L}_{\mathrm{con}}\) on augmentation twins \( (z,z')\) , a VICReg term \( \mathcal{L}_{\mathrm{vic}}\)  [57] with cross-rank all-gather ( var_w \( {=}25\) , cov_w\( {=}1\) , target std \( 1\) ), and a frequency-domain loss \( \mathcal{L}_{\mathrm{freq}}\) given by an \( L_1\) distance between radially weighted log-amplitude FFT spectra of the decoded prediction and the target image. The FFT term is capped to at most two image pairs per batch and only evaluated for \( t<0.7\) so that it improves high-frequency fidelity without destabilising high-noise updates.

Loss-weight schedule. The five \( \lambda\) s are the weightings of the loss components in Eq. (2); they are not tuned independently and none were tuned per experiment. The LossWeightScheduler applies one fixed schedule to every ViT-FMDiT model: (i) \( \lambda_{\mathrm{diff}}\) (\( 0.5{\to}1.0\) ), \( \lambda_{\mathrm{rec}}\) (\( 0.5{\to}0.1\) ), and \( \lambda_{\mathrm{con}}\) (\( 0.3{\to}0.05\) ) are linearly interpolated between fixed endpoints; (ii) the two regularisers are held at constant targets \( \lambda_{\mathrm{vic}}{=}0.05\) and \( \lambda_{\mathrm{freq}}{=}0.30\) but switched on with a delayed onset, i.e. exactly zero for the first epochs and then ramped linearly, so they cannot destabilise the encoder before it has settled. Only four numbers are free per weight (start, end, onset, ramp length); every intermediate value is determined. The remaining constants are not tuned: the VICReg sub-weights are the standard published values, and the frequency-loss sample cap is a memory bound. We use a two-phase encoder schedule: epochs \( 1\) –\( 7\) update only the adaptors and bottleneck, then epoch \( 8\) unfreezes the upper \( 16\) ViT blocks, rebuilds the optimizer, and recomputes the cosine schedule in optimizer-step units. Training runs under PyTorch DDP on \( 4{\times}\) H200 NVL with find_unused_parameters=True to tolerate the topology change; a per-rank skip handshake ensures that NaN-guarded steps are either taken or skipped synchronously across ranks.

Loss-weight sensitivity. We retrained ViT-FMDiT-\( 768\) under deliberate \( \lambda\) perturbations with identical data (a fixed-split subset of \( 39{,}997\) micrographs covering \( 14\) alloys) and identical optimization settings, including a repeat of the reference configuration to measure the run-to-run noise floor, and scored each model on \( 4{,}000\) held-out micrographs (Tab. 6). Every variant converges monotonically (the largest epoch-to-epoch rise in validation loss is \( {<}0.003\) ), and the final validation loss varies by only \( 0.6\%\) across the set. Removing the delayed onset, reducing \( \lambda_{\mathrm{con}}\) six-fold, or changing four weights and both onsets at once shifts PSNR by at most \( 0.09\)  dB and grain disorientation by at most \( 0.21^\circ\) , within a small multiple of the reference-versus-repeat noise floor; no configuration diverges or collapses. In archived runs spanning a \( 75\) -fold range of \( \lambda_{\mathrm{con}}\) , a \( 10\) -fold range of \( \lambda_{\mathrm{diff}}\) , and \( \lambda_{\mathrm{vic}}\in[0,0.2]\) , every configuration likewise converged monotonically.

Bottleneck-capacity ablation (\( d\in{512,768,1024}\) ). Bottleneck width is the most controlled ablation in this family because it leaves the decoder, encoder, conditioning path, and loss schedule unchanged. Under that constraint, held-out ViT-FMDiT PSNR improves monotonically with width, from roughly \( 14.0\) dB at epoch \( 2\) for \( d{=}768\) to roughly \( 16.9\) dB at epoch \( 2\) for \( d{=}1024\) . This is consistent with the main-text choice of \( d{=}512\) as a deliberately severe bottleneck for the latent-BO experiments rather than as the reconstruction-optimal point. Final-epoch reconstruction metrics for all three widths appear in Tab. 1.

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This section expands the codec analysis from Sec. 3.2 and makes explicit the geometric choices that let a generic image decoder serve as a microstructure prior. The thread is simple: every step is chosen to preserve local orientation structure while keeping the RGB image numerically stable for downstream generative models.

Quaternion parameterisation and HCP symmetry. We first convert each per-grain orientation from Bunge–Euler angles (\( ZXZ\) ) to a unit quaternion \( q=(q_x,q_y,q_z,q_w)\in S^3\)  [58]. Because quaternions form a double cover (\( q\equiv -q\) ), we fix the sign by enforcing the hemisphere convention \( q_w\ge 0\) . The relevant crystal symmetry is the HCP point group \( D_6\) with \( |\mathcal{S}_{\mathrm{HCP}}|=12\) , generated by the six \( c\) -axis rotations \( R_z(k\pi/3)\) , \( k=0,…,5\) , and the six compositions \( R_z(k\pi/3)\cdot R_x(\pi)\) . In quaternion coordinates \( R_z(\theta)=(\sin(\theta/2)\,\hat z,\cos(\theta/2))\) under the standard Hamilton product. Two orientations are therefore physically identical whenever they differ by a symmetry element \( s_k \in \mathcal{S}_{\mathrm{HCP}}\) .

Step 1 — continuous unfolding (anchored BFS). Let \( \mathcal{N}=(\mathcal{V},\mathcal{E})\) denote the grain adjacency graph from Dream.3D, with \( |\mathcal{V}|=G\) grains and \( \mathcal{E}\) the set of \( 4\) -connected boundary pairs. We choose the BFS root deterministically as the largest grain,

and align its quaternion to the class anchor (Step 2) by \( q_{g^\star} \leftarrow \arg\max_{s,\sigma}\, \sigma\,\langle s\cdot q_{g^\star},\, \bar{q}_c\rangle\) over \( s\in\mathcal{S}_{\mathrm{HCP}}\) and \( \sigma\in{\pm 1}\) , where \( \langle\cdot,\cdot\rangle\) is the \( \mathbb{R}^4\) inner product. Using the largest grain as the seed shortens the average BFS path length, and aligning that single seed to \( \bar q_c\) keeps the unfolded tree on a common hemisphere so that identical orientations map to consistent colours across the class. Each remaining grain then inherits, from its already visited neighbour \( g_p\) , the symmetry equivalent and sign with maximal inner product:

Each edge requires \( 24\) dot products, so the encode remains \( \mathcal{O}(|\mathcal{E}|)\) in the boundary count. After this pass, quaternion distance between adjacent grains tracks their physical misorientation directly, without the artificial jumps induced by per-grain folding to a fundamental zone.

Step 2 — class anchor via Markley averaging. For each class \( c\) , the anchor \( \bar q_c\) is the \( L_2\) -optimal mean of \( M{=}50\) representative file-level mean quaternions \( {m_j}_{j=1}^M\) . Following [69], it is the leading eigenvector of

The point of using a class-level rather than per-sample anchor is that generated images arrive without per-sample metadata. The cost is a mild long-range colour drift induced by within-class spread, but at \( M{=}50\) that drift is still comfortably inside the capacity of the encoder bottleneck.

Steps 3–4 — frame centring and stereographic projection. Once the anchor is factored out via \( q_g \leftarrow \bar q_c^{-1}\cdot q_g\) , the distribution concentrates near identity (\( q_w\approx 1\) , \( \left\lVert q_{xyz}\right\rVert \ll 1\) ). Enforcing \( q_w\ge 0\) , we project to

which is \( C^\infty\) on the open hemisphere \( q_w>0\) . The inverse map is closed form,

and uses only rational arithmetic. This is the crucial numerical simplification: there is no square root, so decoder noise cannot drive the inverse into NaNs. As \( \left\lVert S\right\rVert\to\infty\) , \( q_w\) approaches \( -1\) smoothly rather than catastrophically.

Step 5 — quantisation error bound. For a centred unit quaternion near identity, let \( S=q_{xyz}/(1+q_w)\) and perturb it by \( \delta S\) . To first order, \( \delta q_{xyz}\approx (1+q_w)\,\delta S\) while \( \delta q_w\) is second order. The induced angular error is therefore \( \delta\theta = 2\arcsin(\left\lVert\delta q_{xyz}\right\rVert) \approx 2(1+q_w)\left\lVert\delta S\right\rVert \approx 2\left\lVert\delta S\right\rVert\) near identity. With quantisation step \( \Delta = 2/(2^b{-}1)\) , this yields the worst-case bound \( \delta\theta \lesssim 0.78^\circ\) for \( b{=}8\) and \( 0.004^\circ\) for \( b{=}16\) . The measured PNG mean in Tab. 8 is \( 0.6^\circ\) , in line with that estimate.

Why these choices. The design logic is now clear. The rational inverse in (8) replaces the unstable recovery \( \sqrt{1-q_x^2-q_y^2-q_z^2}\) , which fails as soon as decoder noise makes the radicand negative. Continuous unfolding preserves the proportionality between RGB distance and true misorientation, keeping the encoded image in the low-frequency regime that pretrained image backbones model well. The class-level Markley anchor is not mathematically ideal, but it is the only option compatible with generated images that lack per-sample metadata. Finally, we use \( b{=}8\) inside Co-PiLOT because pretrained ViT and diffusion backbones expect 8-bit RGB, and the resulting \( \sim 0.6^\circ\) floor remains below both typical EBSD angular resolution and the Read–Shockley low-angle threshold.

This section records the constitutive model and post-processing pipeline actually used by the oracle. The main text only needs the fact that Damask returns a stress–strain response; here we spell out the kinematics, hardening law, and property extraction so the simulation layer is unambiguous.

Kinematics and resolved shear stress. We use the standard multiplicative split \( \mathbf{F}=\mathbf{F}_e\mathbf{F}_p\) , which separates elastic lattice stretch from plastic shear [65]. Plastic flow is written as the sum over all active slip and twin systems, \( \mathbf{L}_p = \sum_\alpha \dot\gamma^\alpha\, \mathbf{s}^\alpha \otimes \mathbf{m}^\alpha\) , with \( \mathbf{s}^\alpha\) and \( \mathbf{m}^\alpha\) the unit slip or twin direction and plane normal. The resolved shear stress on system \( \alpha\) is then \( \tau^\alpha = \boldsymbol{\sigma}:(\mathbf{s}^\alpha\otimes\mathbf{m}^\alpha)\) .

Power-law flow rule. Shear rates follow a phenomenological power law, separately on slip (\( sl\) ) and twin (\( tw\) ) systems:

with reference rates \( \dot\gamma_0^{sl/tw}\) , rate-sensitivity exponents \( n_{sl/tw}\) , and critical resolved shear stresses \( \xi^\alpha_{sl/tw}\) . Twinning contributes only for \( \tau^\alpha>0\) , reflecting the polarity of twin shear in HCP crystals [66].

Hardening evolution. The CRSS evolves through self- and latent-hardening interactions, \( \dot\xi^\alpha = \sum_\beta h^{\alpha\beta}\,|\dot\gamma^\beta|\) , where \( h^{\alpha\beta}\) is the system-interaction matrix. Slip systems use saturation-type hardening with explicit slip–twin coupling,

with initial slip-hardening modulus \( h_0^{sl}\) , saturation CRSS \( \xi^\infty_{sl}\) , hardening exponent \( a_{sl}\) , and slip–twin coupling factor \( f^{sl\text{-}tw}_{sat}\) . Twin systems harden symmetrically through twin–twin and twin–slip channels,

The model is intentionally phenomenological: hardening is encoded through these scalar moduli rather than through dislocation-density or twin-volume-fraction state variables [65]. That choice keeps the oracle tractable while preserving the slip–twin asymmetry that matters most in HCP magnesium [66]. Although damage-augmented variants exist [67], we do not use them here. Instead, every run uses the AZ31-calibrated parameter file distributed with Damask (AZ31_Phenopower.yaml), with basal, prismatic, and pyramidal-\( \langle a\rangle\) /-\( \langle c{+}a\rangle\) slip plus tensile twinning, under isothermal uniaxial tension along the extrusion direction and traction-free transverse faces.

Hollomon fit and property extraction. Post-processing is straightforward but consistent across all runs. We read the HDF5 output, compute volume-averaged Cauchy stress and logarithmic strain along the loading axis at every increment, identify the elastic–plastic transition by the \( 0.2\%\) offset rule, and fit the plastic branch in log–log coordinates with the Hollomon law \( \sigma = K\varepsilon_p^n\) . The yield stress \( \sigma_y\) is the offset intercept, \( \sigma_u\) the maximum stress, and \( (K,n)\) the fitted Hollomon parameters. Runs that do not converge, contain NaN stresses, or produce fewer than \( 50\) converged increments are discarded and logged.

This appendix gathers the implementation detail for the optimizer benchmark that would be distracting in the main text: the pipeline-level adaptations applied uniformly across methods, the concrete baseline settings, the compact design comparison in Tab. 10, and the full hyperparameter table in Tab. 11. All values are those committed in config/az31_*.yaml and copilot/optimizers/ of the released code; no per-decoder, per-objective, or per-seed retuning was introduced after the sweep design was fixed.

F.1 Pipeline-level adaptations applied to the latent-BO optimizers

F.2 Dante [59]

F.3 TuRBO [13]

F.4 BAxUS [14]

F.5 Additional baselines: CMA-ES, DDOM, SEIKO, and DDPO

F.6 Meridian (ours)