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

cs.AISep 29, 2026

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

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

Organizations: Institute of Material and Process Design, Helmholtz-Zentrum Hereon, Germany · Institute of Production Technology and Systems, Leuphana University Lüneburg, Germany · AI for Physical Systems, German Research Center for Artificial Intelligence (DFKI), Germany · Saarland University, Germany

Abstract

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.

Figures & tables

Figure 1: Co-PiLOT . (a) General framework and its materials instantiation (b): a Mg-alloy latent $z^{\star}\in\mathcal{Z}$ is passed through a decoder $\mathcal{D}$ like DiT or FM-DiT , bridged via an orientation codec ( $\Psi$ ) to a .dream3d grid, scored by a crystal plasticity oracle $f$ , and reduced to a fitness $J$ ( $J_{V1}/J_{V2}$ ) that the Meridian optimizer $\mathcal{O}$ feeds back as $z_{\mathbf{new}}$ for the next iteration. Physics enters via the simulator-based objective and feasibility model, and via codec-encoded training data (Sec. 3 ).

Figure 2: (a) Encoder $\mathcal{E}$ , ViT-H/14 trunk $\to$ Attention Pooler with $z\!\in\!\mathbb{R}^{d}$ . (b) Decoder $\mathcal{D}$ , Frozen SD3.5 MMDiT conditioned via a LatentToTokens adaptor and Pooled Projection .

Figure 3: Qualitative Reconstructions on AZ31 and Mg-10Gd. More examples in Fig. 9

No bottleneck
Decoder	$z$	FID $\downarrow$	MS-SSIM $\uparrow$	LPIPS $\downarrow$	$G_{\mathrm{m}}$ $\uparrow$	$\Delta_{\mathrm{ori}}$ $\downarrow$
VQGAN	–	35.63	$0.876\pm 0.12$	$0.132\pm 0.10$	$0.747\pm 0.16$	$27.619\pm 10.01$
Codec	–	0.95	$0.449\pm 0.28$	$0.368\pm 0.15$	$0.926\pm 0.08$	$7.200\pm 4.26$
SDXL	512	108.28	$0.077\pm 0.05$	$0.650\pm 0.08$	$0.370\pm 0.15$	$56.467\pm 8.41$
DiT	512	$\mathbf{23.19}^{*}$	$0.147\pm 0.18$	$0.603\pm 0.08$	$0.486\pm 0.12$	$55.363\pm 10.35$
FM-DiT	512	27.86	$0.169\pm 0.20$	$0.578\pm 0.07$	$\mathbf{0.501\pm 0.11}^{*}$	$\mathbf{53.921\pm 11.68}^{*}$

Table 1: Reconstruction quality and microstructural fidelity on the test set (

n=9{,}084

Figure 4: Best Mean Target-Driven

J_{V2}

versus num. of evaluations (batch size 4,

n_{\mathrm{seed}}=5

Appendix figures & tables12 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 6: Best Mean Toughness-weighted

J_{V1}

versus num. of evaluations (batch size 4,

n_{\mathrm{seed}}=5

Decoder	Batch rule	$J_{V2}\uparrow$	$\sigma_{y}$ (MPa)	$n$	$\sigma_{u}$ (MPa)	evals $\to$ best	collapsed
ViT-FMDiT-768	DPP (ours)	$\mathbf{-0.138\pm 0.007}$	$157.5$	$0.303$	$315.0$	$\mathbf{143}$	$\mathbf{0/3}$
ViT-FMDiT-768	top- $k$	$-0.138\pm 0.010$	$157.6$	$0.303$	$314.7$	$160$	$1/3$
ViT-FMDiT-1024	DPP (ours)	$-0.165\pm 0.010$	$153.6$	$0.305$	$309.7$	$153$	$0/4$
ViT-FMDiT-1024	top- $k$	$-0.162\pm 0.006$	$154.4$	$0.306$	$312.3$	$146$	$0/4$

Appendix

Table 4: Batch-rule ablation (

J_{V2}

160

evaluations). Only the batch selector differs; “collapsed” counts seeds in which the batch degenerated to near-duplicates.

Meridian setting	What differs	$J_{V1}\uparrow$	evals	$J_{V2}\uparrow$	evals
published reference (Alg. 2 )	—	$\mathbf{+0.961\pm 0.009}$	$58$	$\mathbf{-0.136\pm 0.009}$	$106$
earlier trust-region + shell	compact/fixed trust region, DPP pool $64$ , shell $k_{\sigma}\in\{1.0,2.0\}$ , faster/slower shrink	$+0.944\pm 0.003$	$53$	$-0.147\pm 0.005$	$104$
DPP $\to$ greedy top- $k$	diversity term removed	$+0.956\pm 0.012$	$93$	$-0.142\pm 0.013$	$105$
spread (max $-$ min)		$\mathbf{0.017}$		$\mathbf{0.011}$

Appendix

Table 5: Meridian control-parameter sensitivity on ViT-FMDiT-

768

. “evals” = evaluations to plateau.

Figure 8: Companion decoder pipelines (deferred from Fig. 2 ). (a) ViT $+$ DiT-XL/2 ( trainable , $256{\to}512$ ): ClassProj repurposes the DiT class-embedding slot to inject $z$ ; $16{\times}1152$ spatial tokens; SD 1.5 VAE. (b) ViT $+$ SDXL (frozen): $77{\times}2048$ tokens via $3$ AdaLN-Zero blocks (Xavier- $0.10$ ) plus Pooled Projection $\to$ $1280$ -D; SD-XL VAE.

Configuration	$\lambda$ change	Val. loss $\downarrow$	PSNR $\uparrow$	SSIM $\uparrow$	LPIPS $\downarrow$	Disor. ( ∘ ) $\downarrow$
reference	committed schedule	$0.4620$	$12.47$	$0.303$	$0.592$	$59.15$
reference (repeat)	none (noise floor)	$0.4621$	$12.45$	$0.298$	$0.594$	$58.99$
no delayed onset	regularisers from epoch $0$	$0.4595$	$12.39$	$0.285$	$0.587$	$58.94$
weak-reg + late onset	$4$ weights + both onsets	$0.4603$	$12.48$	$0.299$	$0.589$	$59.05$
low $\lambda_{\mathrm{con}}$	$\lambda_{\mathrm{con}}\times 1/6$	$0.4604$	$12.39$	$0.291$	$0.590$	$59.08$
spread (max $-$ min)		$\mathbf{0.0026}$	$\mathbf{0.09}$	$\mathbf{0.018}$	$\mathbf{0.007}$	$\mathbf{0.21}$

Appendix

Table 6: Loss-weight sensitivity of ViT-FMDiT-

768

(

4{,}000

held-out micrographs). The repeated reference measures the noise floor.

Hyperparameter	ViT-FMDiT	ViT-DiT	ViT-SDXL
Backbone	SD3.5 MMDiT ( ${\sim}2.5$ B)	DiT-XL/2 ( ${\sim}0.7$ B)	SDXL UNet ( ${\sim}2.6$ B)
Backbone state	frozen	trainable	frozen (LoRA $r{=}16$ )
VAE	SD3.5 (16ch, $s{=}1.5305$ )	sd-vae-ft-mse (4ch)	SD-XL VAE
Bottleneck $d$ (head./abl.)	$512/\{768,1024\}$	$512/\{768,1024\}$	$512$
LatentToTokens (#q / depth)	$16/5$	$16$ spatial / –	$77/3$
Conditioning dim	$4096$	$1152$	$2048$

Appendix

Table 7: Decoder-specific hyperparameters on

4{\times}

H200. Batch size

=

per-GPU

\times

accum

\times

#GPUs.

Figure 9: Extended reconstruction gallery on five additional alloy classes (rows, in order: ME21 , ZX10 , Mg-5Gd-0.5Mn , Z1 , ZNd10 ); columns follow Fig. 3 .

Class	Boundary F1	Grain-size $r$	PNG-vs-TIFF mean ( ∘ )	PNG-vs-TIFF max ( ∘ )
ME21_extruded	$0.9999$	$0.9997$	$0.61$	$1.35$
Mg-10Gd_extruded	$1.0000$	$0.9993$	$0.69$	$1.30$
AZ31_extruded_HT	$0.9999$	$0.9991$	$0.61$	$1.37$

Appendix

Table 8: Codec round-trip accuracy on three Mg-alloy classes (

300{\times}300

images,

257

–

2{,}044

grains/sample). Boundary F1 and grain-size Pearson correlation are unchanged between TIFF and PNG; only quantisation noise differs.

Parameter	Value
Loading mode	uniaxial tension along ED
Total time $t$ (s)	$250$
Increments $N$	$1250$
Strain rate $\dot{\varepsilon}$ (s -1 )	$1.0\times 10^{-3}$
Output frequency $f_{\mathrm{out}}$	every increment
Constitutive law	phenopowerlaw (HCP)

Appendix

Table 9: Damask simulation parameters used for all property evaluations. The same load.yaml and AZ31_Phenopower.yaml are used across every cell of E2; only the input .dream3d (i.e., the decoded microstructure) varies.

Design axis	Dante [ 61 ]	TuRBO [ 13 ]	BAxUS [ 14 ]	Meridian (ours)
Surrogate	MLP point estimate	ARD-Matérn- $5/2$ GP	ARD-Matérn- $5/2$ GP in subspace	deep-kernel GP
Target-aware signal	scalar- $Y$ only	scalar- $Y$ only	scalar- $Y$ only	property-head EI for $V2$
Feasibility handling	none	none	reject feedback to GP	shared-trunk classifier $g_{\psi}$
Proposal geometry	global tree expansion	isotropic trust region	low-dim subspace trust region	anisotropic trust region
Manifold prior	none	none	PCA-of-seeds embedding	active subspace + adaptive shell
Batch strategy	top- $k$ predicted	max-posterior sampling	max-posterior sampling	quality-weighted DPP

Appendix

Table 10: Key design distinctions among the four latent-optimizers benchmarked on every decoder. Mechanisms unique to Meridian are highlighted in bold. The additional baselines of App. F.5 have no feasibility model, trust region, or manifold prior.

Checklist item	Where addressed	Asset
Code released	Secs. 1 , 6	encoder–decoder: https://github.com/mahishguru/microstructure-encoder-decoder ; orientation codec: https://github.com/mahishguru/orientation-codec ; Meridian and Damask harness: https://github.com/mahishguru/meridian
Decoder hyperparameters	App. C	Tabs. 7 , 6
Simulator hyperparameters	App. E	Tab. 9
Optimizer hyperparameters	App. F	Tab. 11
Sensitivity / ablations	App. B	Tabs. 4 , 5
Compute disclosed	Sec. 4 ; Sec. 5 ; App. A	—

Appendix

Table 12: Reproducibility checklist items to the section, file, or asset that satisfies them.

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Inverse materials design is shifting materials discovery from forward prediction to targeted proposal of candidates that satisfy objectives under physical constraints. Here, we review recent advances in generative crystal structure modeling, multimodal learning, and closed-loop design pipelines for crystalline solids. We survey how modern generators learn chemical-structural priors from large databases to enable controllable sampling of periodic structures, and compare leading model classes including variational autoencoders, normalizing flows, autoregressive formulations, and diffusion models. Particular attention is given to how feasibility constraints and physical priors are enforced across the workflow, through representation choices, training objectives, sampling-time guidance, and post-generation screening and relaxation. We also discuss how multimodal learning fuses diverse materials modalities, including crystal structures, thermodynamic, electronic information, microscopy, spectroscopy, processing context, and scientific text, to construct a more universal, transferable representation of chemical space. In addition, diverse inverse-design strategies are examined, particularly those that integrate conditional generation with latent optimization, Bayesian optimization, reinforcement learning, and active learning. Finally, we highlight recurring failure modes, such as surrogate exploitation, diversity collapse, distribution shift, and the stability-synthesizability gap, and outline discovery-grade evaluation practices based on staged reporting of validity, novelty, uniqueness, stability, and cost.

Abstract

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