GUIDe: Generative and Uncertainty-Informed Inverse Design for On-Demand Nonlinear Functional Responses

TL;DR

GUIDe introduces a forward-model-based inverse design framework that leverages a probabilistic surrogate and MCMC sampling to generate designs achieving nonlinear functional responses with quantified uncertainty. By predicting response distributions and computing a tolerance-aware likelihood, GUIDe avoids unreliable inverse mappings and provides diverse, high-likelihood design candidates, including robust extrapolations beyond the training data. Applied to nacre-inspired composites, the method demonstrates strong alignment between forward-predicted likelihood and actual feasibility (r ≈ 0.95) and uncovers mechanics insights about interface-normal dominance and cohesive-energy constraints. The approach offers reliable, data-efficient design exploration, with potential to guide active learning, physics-informed extensions, and broader inverse-design tasks across engineering domains.

Abstract

Inverse design is a common yet challenging engineering problem, particularly for nonlinear functional responses such as mechanical behavior or spectral analysis. Deep generative models are motivated by intractability, non-existence or non-uniqueness of solutions, and the need for rapid solution-space exploration. In this study, we show that deep generative model-based and optimization-based approaches can provide incomplete solutions or hallucinate given out-of-distribution targets. To address this, we propose the Generative and Uncertainty-informed Inverse Design (GUIDe) framework, which leverages probabilistic machine learning, statistical inference, and Markov chain Monte Carlo to generate designs with targeted nonlinear behaviors. Instead of inverse mappings, i.e., response $\mapsto$ design, GUIDe adopts design $\mapsto$ response: a forward model predicts each design's nonlinear functional response and evaluates the confidence under a user-specified tolerance. Sampling the solution space by this confidence yields diverse feasible designs. Our validation on nacre-inspired materials finds solutions beyond the training range, even under out-of-distribution targets.

Figures (6)

• Figure 1: a, GUIDe applies to various inverse design problems where the targets are nonlinear functional responses. b, c, a schematic comparison between inverse-model-based and forward-model-based (GUIDe) approaches. In both cases, the machine learning models are trained on uniformly sampled designs, with mappings between design and response spaces shown as arrows, depicting a scenario in which both approaches conditioned the same OOD target. b, Inverse-model-based algorithms (e.g., tandem neural network and conditional generative models) directly map responses to designs, but yield unreliable solutions for OOD targets. c, Forward-model-based approaches predict responses with uncertainty, then the solutions are sampled based on their likelihood of meeting the target, highlighting the improved trustworthiness that distinguishes the proposed method. d-f, Architecture of GUIDe: d, A data-driven surrogate with uncertainty quantification is first trained to model the responses. e, Given a target response and the tolerance, stochastic optimization methods are applied to approach the feasible region by identifying an initial design. f, The design candidates are generated by sampling the posterior of the design parameters through MCMC, where the likelihood is approximated by integrating the predictive response space over the tolerance bound.
• Figure 2: The design results obtained from the conditional diffusion model given OOD targets (i.e., stress–strain curves represented by a parabola and by Gaussian noise). The results reveal that CDM can generate irrelevant solutions (detailed in Sec. \ref{['subsubsec2']}). On the other hand, GUIDe identified low likelihood of feasible solutions and avoided sampling misleading designs.
• Figure 3: a, Schematic illustration showing the inside view of a red abalone shell. Inset shows a scanning electron microscopic top view revealing a Voronoi patterned brick-and-mortar structure (adapted with permission from ref.barthelat2007mechanics, Elsevier Ltd.). A CAD model for a nacre-inspired composite is built to mimic this structure. b, Trilinear interface laws for normal and shear traction--separation relationship. Ranges of parameters in our dataset are marked by gray bars. c, Stress--strain curves from the dataset, computed via FEM under tensile loading. Representative response behaviors are labeled in red dashed lines. d, Distribution of strength, toughness, Young's modulus, and failure tensile strain for the stress--strain curves in c. e, Evaluation of the correlation between likelihood and actual feasibility rate. All samples were divided into 20 likelihood intervals, each containing at least 20 samples. The sample density is indicated by the blue dashed curve. A strong positive correlation can be observed between the likelihood score and the feasibility rate. f, Comparison of the feasibility rate distributions achieved by GUIDe, diffusion model, and genetic algorithm over 50 design scenarios. For each scenario, the target was sampled at random from the test dataset, and feasibility was computed from 50 generated designs. g, Inverse design results from GUIDe for three representative scenarios out of 50. For each scenario, two examples of the resulting fracture behavior are displayed.
• Figure 4: Design scenarios with OOD targets in the linear regime. a, b, Two design scenarios characterizing ultra-stiff (left) and ultra-soft (right) targets. Stress--strain curves (computed by FEM) for generated feasible designs are shown in comparison with the target, shaded tolerance range, and the closest match in the training data set. By adjusting the tolerance at different strain stages, the constraint was relaxed after the elastic regime; therefore, the model is free to explore designs with various post-yielding behavior. c, d, The comparison between the standardized range of training data and the feasible samples from GUIDe for design scenarios in a and b, respectively. It shows that GUIDe explores well beyond the bounds of the training data and still locates valid solutions.
• Figure 5: Design scenario with an OOD target featuring extremely strong and brittle mechanical behavior. a, Target response, closest matching solution in training data, and the feasible responses of designs generated by GUIDe. The target exhibits a tensile strength of 362 MPa, Young’s modulus of 77.0 GPa, and toughness of 2.82 MPa, while tolerance is as 8.3% of $\sigma_\text{max}$. b, Design sample distribution of GUIDe, CDM, and GA, conditioned on the target introduced in a. GUIDe achieves the broadest exploration among the three. c, Quantitative evaluation of the feasible designs generated by the three methods in metrics of feasibility rate, Vendi score (diversity), and k-NN novelty. While CDM reaches a similar level of feasibility as GUIDe, the latter demonstrates superior performance in generating diverse and novel feasible designs. d, Standardized range of each design dimension across feasible designs obtained by the three methods. e, Feasible designs' distribution shown in pairwise scatter plots of each design dimension. GUIDe exhibits the widest coverage across the design space, often extending far beyond the training distribution. CDM samples are concentrated around the training prior, while GA demonstrates moderate exploration but identifies only three feasible designs. f, Distribution of interface-mode ratio $\phi$ versus applied strain from feasible designs generated by GUIDe. g, Distribution of frequencies across the damage variables of cohesive elements from feasible designs by GUIDe. h, Fracture patterns of four example feasible designs by GUIDe.