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.

翻译：逆向设计问题在工程与材料科学中普遍存在。正向过程（即根据设计参数计算输出量）通常需要运行数值模拟（如有限元法）作为中间步骤，这本身就是一个优化问题。在许多场景中，多个设计参数可能产生相同或相似的输出值。针对此类情况，多模态概率方法在获取多样化解决方案方面具有优势。逆向设计的主要困难源于设计空间的结构，因为离散参数或额外约束会阻碍基于梯度的优化方法的直接应用。为解决这一问题，我们提出了一种基于扩散模型的新型逆向设计方法。我们的方法将原始设计空间松弛为连续网格表示，其中梯度可通过正向模拟中的隐式微分计算。在此松弛参数空间上训练扩散模型，以作为合理松弛设计的先验。通过引导扩散进行参数采样，所用梯度在推理时通过可微分模拟从指定目标函数传播而来。设计样本通过反向投影到原始参数空间获得。我们针对复合材料设计问题开发了该方法，其中正向过程建模为线性有限元问题。我们评估了该方法在匹配指定体积模量的设计中的性能。实验表明，在二维和三维设置中，对于中高目标体积模量，我们的方法能够在1%相对误差范围内提出多样化设计方案。我们还证明，通过使用多目标损失函数，可以同时最小化生成样本的材料密度。