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Co-PiLOT: Constrained Physics-Informed Latent Optimization for Target-Driven Inverse Design

Mahish K. Guru, Mayank Nagar, Ayush vyas, Jan Bohlen, Roland Aydin, Noomane Ben Khalifa

arXiv:2609.37875v1cs.AIcs.CE

TL;DR

Inverse design must search structured, often infeasible designs against costly simulators, complicating direct search. CO-PiLOT uses a generative latent space with a physics-informed optimizer; in Mg-alloy design, ViT-FMDiT-768 with MERIDIAN achieves the strongest target-driven and toughness-weighted objectives against seven baselines.

  • Problem

    Inverse design requires finding feasible structured designs whose expensive simulator outputs match target properties, while raw spaces are non-Euclidean and difficult to search.

    Source: §1 · pp. 1–2

  • Method

    CO-PiLOT couples a pretrained generative decoder and orientation codec to a physics-informed latent optimizer that models feasibility and searches decoder-valid microstructures.

    Source: §1 · p. 2, §3 · p. 4

  • Results

    ViT-FMDiT-768 with MERIDIAN reaches the best target-driven and toughness-weighted objectives against seven baselines under a matched budget.

    Source: §1 · p. 2

  • Takeaways & Limitations

    The orientation codec makes image-domain generators usable inside a crystal-plasticity loop, while MERIDIAN combines uncertainty, feasibility modeling, manifold-aware proposals, and diverse batches to reduce wasted simulator calls.

    Source: §6 · p. 10

    The decoder restricts search to microstructures its learned prior can express, and the paper does not show transfer beyond the studied AZ31 setting or validate designs experimentally.

    Source: §6 · p. 10

Abstract

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Inverse design of physical systems (molecules, devices, microstructures) often reduces to optimizing a high-dimensional structure against an expensive black-box simulator. Direct search is difficult because the space is non-Euclidean, feasibility is hard to encode, and each evaluation is expensive. We present Co-PiLOT, a latent optimization approach that maps candidates through a generative encoder-decoder, uses the decoder as a learned validity prior, and searches the latent space with physics-informed black-box optimization. The framework is applied on the inverse design of magnesium alloy microstructure/texture. We develop a vision transformer based-encoder; paired with latent diffusion, diffusion transformer and rectified-flow transformer-based decoders on $\sim80{,}000$ EBSD-derived microstructure dataset to learn a minimal bottleneck, $z$. The ViT-FMDiT model ($z$=$768$) reconstructs high-fidelity microstructure images (FID $27.86$, MS-SSIM $0.178$), which our self-segmenting orientation codec converts into input grids for crystal plasticity solver. Finally, we introduce MERIDIAN, an active latent optimizer driven by deep-kernel Gaussian-process uncertainty, failure-aware feasibility prediction, manifold-aware trust regions, and target-aware acquisition. Within a budget of $160$ simulations, the ViT-FMDiT and MERIDIAN combination yields the best target-driven objective score, reducing the relative target error by $3$--$22\%$ against seven baselines (DANTE, TuRBO, BAxUS, CMA-ES, DDOM, SEIKO, DDPO) on the same decoder.

1 Introduction

The introduction frames inverse design as expensive search over structured, often infeasible physical designs, then proposes a generative latent framework with an optimizer tailored to simulator failures and material validity.

1 Introduction

  • Raw-space search is difficult because designs are discrete and non-Euclidean, most are physically invalid, and each simulator evaluation is expensive.The introduction describes budgets of 10^2–10^3 evaluations, insufficient for brute-force sampling or high-dimensional Bayesian optimization over raw designs.Source: §1 · p. 1
  • Prior latent optimization approaches assume cheap oracles, while existing crystal models target sampling rather than reliable search with an expensive, occasionally divergent simulator.The motivating Mg-alloy task also involves orientation-dependent plasticity and simulations that can diverge.Source: §1 · p. 2
  • MERIDIAN combines failure-aware feasibility learning with manifold-aware local search, including a latent-shell projection that keeps candidates where the decoder yields valid microstructures.The introduction also identifies CO-PiLOT as the first closed loop coupling an image-domain generative prior to a crystal-plasticity solver under a 160-simulation budget.Source: §1 · p. 2
  • ViT-FMDiT-768 with MERIDIAN achieves the best target-driven and toughness-weighted objectives against seven baselines under a matched budget.The framework combines a decoder family, orientation codec, and latent optimizer for Mg-alloy microstructures.Source: §1 · p. 2
  • CO-PiLOT formulates inverse design as a swappable decoder, oracle, objective, constraint set, and optimizer, and demonstrates the abstraction on Mg-alloy microstructures.The shared ViT encoder and FMDiT decoder are trained on 81,756 EBSD-derived microstructures across 17 Mg-alloy classes; the orientation codec achieves approximately 0.7° round-trip misorientation, below the 5° grain-boundary threshold.Source: §1 · p. 2
  • The framework is demonstrated only on Mg-alloy microstructures, although the authors describe other solver-based applications as possible.The general abstraction is presented as applicable in principle beyond this domain, with other instantiations left to future work.Source: §1 · pp. 2–3

2 Related work

Related work spans latent optimization, generative materials design, and optimization around generators, but prior approaches commonly rely on cheaper predictors or do not address simulator divergence and structured feasibility.

High-dimensional latent optimization

  • Latent-BO methods add weighted retraining, trust regions, and constrained or multi-objective variants, but are largely evaluated with cheap neural property predictors [28] [29] [30].High-dimensional methods such as TURBO, BAXUS, and SAASBO improve sample efficiency without generative priors or explicit treatment of simulator divergence and structured feasibility [13] [14] [31].Source: §2 · p. 3

Generative inverse design of materials

  • Materials inverse-design work includes generative models coupled to regressors or Bayesian optimization, while polycrystal design still relies on informative low-dimensional priors or conditional flows in active-learning loops [16] [41].Crystal generators such as CDVAE, DiffCSP, and MatterGen primarily target sampling or one-shot conditional generation [6].Source: §2 · pp. 3–4

Black-box optimization around generators

  • Generator-centered black-box optimization has been applied to molecules, photonics, and topology optimization, but typically uses cheaper solvers, smoother objectives, or direct conditional generation [28] [34] [35].Diffusion black-box optimization and reward fine-tuning methods assume offline data or substantially larger query budgets [43].Source: §2 · p. 4
  • CO-PiLOT targets constrained latent optimization with a slow, occasionally divergent physics solver, emphasizing calibrated surrogates, feasibility modeling, and robust search in pretrained latent spaces.The related-work discussion identifies this combination of conditions as the gap from prior generator-based approaches.Source: §2 · p. 4

3 Method

CO-PILOT couples a pretrained latent decoder, an orientation codec, crystal-plasticity simulation, and a physics-informed optimizer to search feasible microstructures against target properties. Its formulation separates decoder, oracle, objective, constraints, and optimizer, while MERIDIAN adapts acquisition and search to failures and latent-manifold geometry.

3 Method

  • CO-PILOT couples a pretrained decoder and physics-informed optimizer through an orientation codec, iteratively proposing latent vectors that an oracle simulator scores.The formulation separates decoder, oracle, objective, constraints, and optimizer; decoder outputs lie on a learned manifold within a latent box.Source: §3 · p. 4

3.1 Implementation of Encoder–Decoder (D)

  • A shared ViT-H/14 encoder feeds bottleneck-conditioned diffusion decoders, trained jointly with generative, reconstruction, alignment, and frequency losses.The encoder uses an attention pooler over image tokens; the decoder family includes frozen FM-DiT and SDXL backbones and a trainable DiT.Source: §3.1 · pp. 4–5

3.2 Orientation codec (Ψ)

  • The orientation codec maps HCP orientation fields to RGB images and back, enabling image-domain decoders to produce solver-compatible microstructures.Its encoding reduces discontinuities from symmetry folding and uses stereographic projection; 8-bit quantization has about 0.6° mean error and self-segmenting decode recovers grains without labels.Source: §3.2 · p. 6

3.3 Crystal plasticity simulation pipeline and objectives (f ◦D, J)

  • DAMASK evaluates decoded DREAM.3D polycrystals under AZ31-calibrated uniaxial tension, while objectives score either target-property distance or toughness-weighted strength with grain-count penalties.Non-converged runs, primarily from degenerate decoded grains, are excluded from surrogate training; the toughness-weighted objective also enforces a hardening floor.Source: §3.3 · pp. 6–7

3.4 Optimizer: MERIDIAN (Ours) (O)

  • MERIDIAN combines a feasibility classifier and GP uncertainty with shell-projected, importance-weighted trust-region search and diverse, target-aware batch acquisition.It fits the GP on feasible observations, gates qLogNoisyExpectedImprovement by predicted feasibility, and restarts when the trust region collapses or progress plateaus.Source: §3.4 · pp. 7–8

4 Experiment 1: Microstructure reconstruction (E1)

The reconstruction experiment compares decoder fidelity on a held-out set of EBSD-derived microstructures. ViT-DiT leads in FID, while ViT-FMDiT-768 leads bottlenecked decoders on grain recovery and several other fidelity metrics; its round-trip orientation reshuffle preserves class statistics rather than producing physical violations.

4 Experiment 1: Microstructure reconstruction (E1)

  • The downstream models use frozen encoder–decoder checkpoints trained on 81,756 microstructures, with 9,084 held out for testing.The corpus was expanded through geometry- and physics-aware oversampling.Source: §4 · p. 8
  • ViT-DiT achieves the best FID, while ViT-FMDiT-768 leads bottlenecked decoders on Gm, MS-SSIM, LPIPS, and Δori.The table reports reconstruction quality and microstructural fidelity on 9,084 test examples.Source: §4 · p. 8, Table 1 · p. 9
  • AZ31 and Mg-10Gd reconstructions preserve macroscopic morphology and class statistics, while ViT-SDXL produces visibly washed-out grains.The qualitative reconstructions are shown in Figure 3; ViT-SDXL also has the worst reported metrics, including FID ∼108.Source: §4 · p. 8, Figure 3 · p. 8
  • The bottlenecked decoders introduce about 53° per-pixel round-trip disorientation, a class-preserving reshuffle rather than a physical violation.Decoded grains remain in the HCP fundamental zone, and their texture statistics match the target class.Source: §4 · pp. 8–9

5 Experiment 2: Materials inverse design (E2)

E2 tests whether the optimizer suite and trained decoders can find statistically valid AZ31 microstructures matching a specified mechanical target; ViT-FMDiT-768 with MERIDIAN gives the strongest target-driven configuration.

5 Experiment 2: Materials inverse design (E2)

  • Each run receives 160 new simulator evaluations, and the target-driven comparison uses five independent runs per cell.The runs use a shared 100-evaluation seed cache for warm starts; the cache is excluded from the new-evaluation budget.Source: §5 · p. 9
  • Across ViT-FMDiT-768/1024, MERIDIAN remains best; DDOM is within one standard deviation but never ahead, while CMA-ES trails and SEIKO and DDPO do not beat the warm-start incumbent within 160 evaluations.The middle of the ranking reorders between decoders while the top two do not.Source: §5 · p. 9
  • After 144 evaluations, ViT-FMDiT-768 with MERIDIAN yields the closest overall target match, JV 2 = −0.132±0.004, with σy = 159.1 MPa, n = 0.299, and σu = 314.0 MPa.The result falls within the AZ31 material class.Source: §5 · p. 9, Table 2 · p. 10

6 Conclusion

CO-PILOT couples a learned microstructure prior to physics-informed optimization, with MERIDIAN giving the strongest AZ31 target-driven result and effectiveness on the toughness objective. The demonstrated scope remains bounded by the decoder, simulator setting, and evaluation budget.

6 Conclusion

  • CO-PILOT can search only microstructures expressible by its decoder, which is a class-conditional generator of statistically valid AZ31 microstructures rather than a faithful per-pixel orientation reconstructor.The encoder–decoder remains non-isometric in crystallographic rotation distance despite the codec's subdegree accuracy.Source: §6 · p. 10
  • The DAMASK oracle covers HCP phenopower-law plasticity without damage, under isothermal loading and the loading paths studied; inverse-designed microstructures require multi-loading and experimental validation.The paper does not show transfer to other polycrystalline materials, property sets, or physical domains, and compares optimizers at one 160-evaluation budget.Source: §6 · p. 10
  • On AZ31 design, ViT-FMDiT-768 with MERIDIAN gives the strongest target-driven result and remains effective on the toughness objective.The framework couples learned generative design spaces to simulator-driven inverse design; transfer to other physical systems remains to be shown.Source: §6 · p. 10

A Extended E2 results

For the toughness-weighted objective, ViT-FMDiT-768 with MERIDIAN remains the most effective pairing, while the wider 1024-D latent exposes texture pockets that are difficult to model with limited simulator calls.

A Extended E2 results

  • ViT-FMDiT-768 with MERIDIAN reaches JV 1 = 0.961 ± 0.008, with σy = 157.9 MPa, n = 0.305, σu = 316.8 MPa, and 112 optimizer-phase simulations to the best value.This is the strongest overall pairing for the toughness-weighted objective.Source: Table 3 · p. 16, Figure 6 · p. 16, A Extended E2 results · p. 16
  • The toughness-weighted objective seeks high-strength, high-plastic-work microstructures without falling below the hardening floor; the best solution shifts slightly toward higher ultimate strength while preserving hardening.The protocol uses 160 optimizer-phase simulator evaluations per run.Source: A Extended E2 results · p. 16
  • In the 1024-D latent, isolated texture pockets are difficult to model globally from 160 calls, while sparse local perturbations and PCA-aligned subspace moves can exploit them.Across both objectives, ViT-FMDiT-768 is the best operating point: enough capacity for useful AZ31 textures without losing sample efficiency.Source: A Extended E2 results · pp. 16–17

B Additional ablations and sensitivity of MERIDIAN

MERIDIAN's ranking holds across decoders and objectives, while ablations indicate that batch diversity improves efficiency and robustness rather than peak quality; control-parameter changes have limited effects.

B Additional ablations and sensitivity of MERIDIAN

  • MERIDIAN has the best mean objective in all eight studies, and ViT-FMDiT-768 is the best decoder in all eight optimizer–objective combinations.Under a uniformly random ranking among four optimizers on every decoder, the probability of MERIDIAN's ranking is (1/4)^8 ≈ 1.5×10^-5.Source: B Additional ablations and sensitivity… · p. 17
  • On ViT-FMDiT-768, DPP and top-k reach the same mean quality, JV 2 = −0.138, but DPP reaches the plateau about 11% earlier, with lower spread and no batch-collapse failure.On ViT-FMDiT-1024 the rules tie statistically, −0.165±0.010 vs. −0.162±0.006, with no collapsed seeds; DPP's benefit is sample efficiency and robustness rather than peak performance.Source: Table 4 · p. 17, B Additional ablations and sensitivity… · p. 17
  • Under JV 2, every tested MERIDIAN control setting reaches its plateau at 104–106 of 160 evaluations, with no stalls, divergence, or convergence failures.The achieved optimum moves by about one seed-level standard deviation; the published setting is best on both objectives despite being tuned only on JV 1.Source: Table 5 · p. 17, B Additional ablations and sensitivity… · p. 17

C Encoder–decoder details

This section documents the shared encoder, three decoder-conditioning interfaces, composite training objective, and selected sensitivity and bottleneck-width results.

C Encoder–decoder details

  • OpenCLIP ViT-H/14 supplies the shared encoder, with its lower 16 blocks frozen, upper 16 trainable, and a four-query attention pooler feeding bottlenecks d ∈ {512, 768, 1024}.The encoder processes 512×512 images and uses the pretrained laion2b_s32b_b79k weights.Source: C Encoder–decoder details · p. 18
  • The three decoder interfaces differ: FM-DiT and SDXL use learned token adaptors, while DiT-XL/2 uses class-slot and spatial-token conditioning.The companion pipelines use trainable DiT-XL/2 with the SD 1.5 VAE and frozen SDXL with the SD-XL VAE.Source: C Encoder–decoder details · p. 18, Figure 8 · p. 19
  • The five-term training objective combines diffusion, clean-latent reconstruction, contrastive alignment, VICReg regularisation, and a gated frequency-domain loss.The frequency loss is evaluated on a small subsample at t < 0.7; the fixed weight schedule delays the regularisers’ onset.Source: C Encoder–decoder details · pp. 18–19
  • Weight perturbations changed PSNR by at most 0.09 dB and grain disorientation by at most 0.21°; all tested variants converged without collapse.The reported shifts were within a small multiple of the reference-versus-repeat noise floor.Source: C Encoder–decoder details · p. 19, Table 6 · p. 20
  • At epoch 2, held-out ViT-FMDiT PSNR rose from roughly 14.0 dB at d=768 to roughly 16.9 dB at d=1024 as width increased.The passage frames d=512 as a deliberately severe bottleneck for latent-BO experiments, not the reconstruction-optimal width.Source: C Encoder–decoder details · p. 20

D Orientation codec — extended theory and round-trip diagnostics

The codec represents HCP orientations as RGB while preserving local orientation relationships and keeping the image numerically stable for generative models.

D Orientation codec — extended theory and round-trip diagnostics

  • HCP orientations are represented by unit quaternions, with sign fixed to qw ≥ 0 and physically equivalent orientations identified through the 12-element D6 symmetry group.The group comprises six c-axis rotations and six compositions of those rotations with Rx(π).Source: D Orientation codec — extended theory… · p. 20
  • Figure 9 extends the reconstruction gallery to ME21, ZX10, Mg-5Gd-0.5Mn, Z1, and ZNd10, with columns following Figure 3.The caption identifies the gallery layout but does not state a comparative outcome.Source: Figure 9 · p. 21
  • A breadth-first traversal replaces each grain quaternion with the symmetry-equivalent orientation closest to its visited parent, so adjacent-grain quaternion distance tracks physical misorientation.The largest grain is chosen as the deterministic BFS root; the traversal takes O(|E|) in the boundary count.Source: D Orientation codec — extended theory… · pp. 20–21
  • A class-level Markley anchor aligns the unfolded orientations; using one anchor per class accommodates generated images without per-sample metadata, at the cost of mild long-range colour drift.The anchor is calculated from M=50 representative file-level mean quaternions.Source: D Orientation codec — extended theory… · p. 21
  • The rational inverse stereographic map avoids square roots, so decoder noise cannot make orientation recovery fail through a negative radicand or NaNs.The reported map is C∞ on the open hemisphere qw > 0.Source: D Orientation codec — extended theory… · pp. 21–22
  • With 8-bit quantisation, the stated near-identity worst-case angular-error bound is 0.78°; the measured PNG mean is 0.6°.The paper uses 8-bit RGB because pretrained ViT and diffusion backbones expect it, and reports the resulting floor below typical EBSD angular resolution and the Read–Shockley low-angle threshold.Source: D Orientation codec — extended theory… · p. 22

E DAMASK and the HCP phenopower-law constitutive model (Fp, ˙γα, ξα)

This section specifies the DAMASK constitutive model, loading conditions, and post-processing used to extract properties from simulated microstructures.

E DAMASK and the HCP phenopower-law constitutive model (Fp, ˙γα, ξα)

  • The model uses the multiplicative split F = FeFp and represents plastic flow as the sum of shear on active slip and twin systems.Resolved shear stress is computed by contracting the Cauchy stress with each system’s direction and plane normal.Source: E DAMASK and the HCP phenopower-law… · p. 22
  • Slip and twin shear rates follow separate phenomenological power laws, and twinning contributes only when resolved shear stress is positive.This encodes the polarity of twin shear in HCP crystals.Source: E DAMASK and the HCP phenopower-law… · p. 22
  • Critical resolved shear stresses evolve through self- and latent-hardening interactions, with saturation-type slip hardening and explicit slip–twin and twin–twin coupling.Twin systems harden symmetrically through twin–twin and twin–slip channels.Source: E DAMASK and the HCP phenopower-law… · p. 22
  • The intentionally phenomenological hardening law uses scalar moduli rather than dislocation-density or twin-volume-fraction state variables; runs use AZ31-calibrated parameters under isothermal uniaxial extrusion-direction tension.The setup includes basal, prismatic, pyramidal slip and tensile twinning, with traction-free transverse faces and no damage-augmented variant.Source: E DAMASK and the HCP phenopower-law… · pp. 22–23
  • Post-processing extracts yield stress by the 0.2% offset rule, maximum stress, and Hollomon parameters; nonconverged runs, NaN stresses, or runs with fewer than 50 converged increments are discarded.The same DAMASK parameter files are used across every E2 cell; only the input microstructure varies.Source: E DAMASK and the HCP phenopower-law… · p. 23, Table 9 · p. 23

F Baseline optimizer implementations and hyperparameters

The appendix documents shared pipeline adaptations, baseline configurations, and the implementation choices used to run the latent-optimizer comparison under a fixed simulation budget.

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

  • Failed decode, codec, and solver evaluations are excluded from regression fitting across the four latent-BO optimizers, while MERIDIAN additionally retains them for feasibility learning.A shared failure cap of 8 makes trust-region restarts reachable within the 200-evaluation regime described for these adaptations.Source: §F.1 · p. 23

F.2 DANTE [61]

  • DANTE uses Neural Tree Explorer with a deep MLP surrogate and iterative root-and-cloud DUCB tree search, without a restart mechanism.Its implementation expands 200 leaves per round with branching factor 16 and depth-decayed Gaussian leaves.Source: §F.2 · p. 24, DANTE · p. 25

F.3 TURBO [13]

  • TURBO uses an ARD-Matérn-5/2 GP with Thompson sampling and a trust region; a five-batch plateau guardrail supplements its capped failure rule.The added restart trigger addresses the limited 40-iteration budget.Source: §F.3 · p. 24, TURBO · p. 25

F.4 BAXUS [14]

  • BAXUS fits a GP in a low-dimensional PCA-of-seeds subspace and sends soft-floor codec failures back to the GP as reject feedback.Its embedding uses leading PCA directions of the class-conditional latent pool.Source: §F.4 · p. 24, BAXUS · p. 25

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

  • CMA-ES and DDOM are optimizer-level comparisons, while SEIKO and DDPO modify the decoder; DDPO reference configurations use 25,600–57,600 reward queries, 160–360× the budget.The appendix describes how each method is adapted or differs from MERIDIAN in the fixed-budget loop.Source: §F.5 · p. 24

F.6 MERIDIAN (ours)

  • MERIDIAN jointly trains a deep-kernel feature trunk and feasibility classifier, then fits its property head on feasible samples for target-driven objectives.The heteroscedastic head predicts (σy, σu, n, K, ngrains), and its linear layer is reset every 10 refits to avoid variance collapse.Source: §F.6 · p. 24, MERIDIAN · p. 25
  • MERIDIAN shapes proposals using active-subspace anisotropy, a shell-projected incumbent region, sparse and diffuse moves, and a small off-shell exempt fraction.After round 35 of 40, it switches to local polishing; restarts follow trust-region collapse or five plateau rounds, with termination after three unsuccessful restarts.Source: MERIDIAN · p. 26
  • A greedy DPP selects each batch from the top 256 acquisition candidates to promote diversity in feature space while retaining latent-space separation.The design comparison table distinguishes the mechanisms of the four latent optimizers; additional baselines lack feasibility models, trust regions, and manifold priors.Source: Table 10 · p. 25, MERIDIAN · p. 26

G Reproducibility checklist mapping

Table 12 maps reproducibility checklist items to the section, file, or asset that addresses each item.

G Reproducibility checklist mapping

  • Table 12 maps reproducibility checklist items to the section, file, or asset that satisfies them.Source: Table 12 · p. 26

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