Guided Diffusion by Optimized Loss Functions on Relaxed Parameters for Inverse Material Design

Abstract

Inverse design problems are common in engineering and materials science. The forward direction, i.e., computing output quantities from design parameters, typically requires running a numerical simulation, such as a FEM, as an intermediate step, which is an optimization problem by itself. In many scenarios, several design parameters can lead to the same or similar output values. For such cases, multi-modal probabilistic approaches are advantageous to obtain diverse solutions. A major difficulty in inverse design stems from the structure of the design space, since discrete parameters or further constraints disallow the direct use of gradient-based optimization. To tackle this problem, we propose a novel inverse design method based on diffusion models. Our approach relaxes the original design space into a continuous grid representation, where gradients can be computed by implicit differentiation in the forward simulation. A diffusion model is trained on this relaxed parameter space in order to serve as a prior for plausible relaxed designs. Parameters are sampled by guided diffusion using gradients that are propagated from an objective function specified at inference time through the differentiable simulation. A design sample is obtained by backprojection into the original parameter space. We develop our approach for a composite material design problem where the forward process is modeled as a linear FEM problem. We evaluate the performance of our approach in finding designs that match a specified bulk modulus. We demonstrate that our method can propose diverse designs within 1% relative error margin from medium to high target bulk moduli in 2D and 3D settings. We also demonstrate that the material density of generated samples can be minimized simultaneously by using a multi-objective loss function.

Figures (11)

• Figure 1: Overview of our proposed method. Original design parameters are relaxed into a continuous grid representation on which a diffusion model acts as prior. We guide the sample generation process zero-shot by objective functions using implicit differentiation of the FEM solver. Our approach finds diverse samples with low objective cost.
• Figure 2: Inverse 2D material designs. Generated samples for selected bulk moduli $K^*$, ordered by relative error quantile. Best on the left, worst on the right. Labels show $K_s \, / \, K_\theta \, / \, \epsilon_r$. The values $(E,\nu,\rho)$ in the normalized coordinate space are encoded as $(r,g,b)$ values of the image. Our model is able to propose diverse and plausible designs close to the target bulk moduli.
• Figure 3: Inverse 3D material designs. Generated samples for selected bulk moduli $K^*$, ordered by relative error quantile. Best on the left, worst on the right. Labels show $K_s \, / \, K_\theta \, / \, \epsilon_r$. The values $(E,\nu,\rho)$ in the normalized coordinate space are encoded as $(r,g,b)$ values of the image. Our model is able to propose diverse and plausible designs close to the target bulk moduli.
• Figure 4: Visualization of base materials used. Color represents the index in the 168 non-empty chunks.
• Figure 5: Histograms of bulk modulus $K$ of individual examples in datasets. Guidance targets are chosen uniformly spaced between the 1 and 99 percentile.