A Score-based Generative Solver for PDE-constrained Inverse Problems with Complex Priors

TL;DR

This paper introduces a score-based diffusion approach combined with physics-informed neural network surrogates to efficiently solve high-dimensional PDE-constrained inverse problems with complex priors, outperforming existing methods.

Contribution

It proposes a novel score-based generative sampling method integrated with CNN surrogates for PDE inverse problems with complex priors, improving efficiency and accuracy.

Findings

Effective learning of geometrical features from prior samples

Enhanced inverse estimation accuracy over state-of-the-art methods

Successful application to hyper-elastic and multi-scale mechanics problems

Abstract

In the field of inverse estimation for systems modeled by partial differential equations (PDEs), challenges arise when estimating high- (or even infinite-) dimensional parameters. Typically, the ill-posed nature of such problems necessitates leveraging prior information to achieve well-posedness. In most existing inverse solvers, the prior distribution is assumed to be of either Gaussian or Laplace form which, in many practical scenarios, is an oversimplification. In case the prior is complex and the likelihood model is computationally expensive (e.g., due to expensive forward models), drawing the sample from such posteriors can be computationally intractable, especially when the unknown parameter is high-dimensional. In this work, to sample efficiently, we propose a score-based diffusion model, which combines a score-based generative sampling tool with a noising and denoising process driven by stochastic differential equations. This tool is used for iterative sample generation in accordance with the posterior distribution, while simultaneously learning and leveraging the underlying information and constraints inherent in the given complex prior. A time-varying time schedule is proposed to adapt the method for posterior sampling. To expedite the simulation of non-parameterized PDEs and enhance the generalization capacity, we introduce a physics-informed convolutional neural network (CNN) surrogate for the forward model. Finally, numerical experiments, including a hyper-elastic problem and a multi-scale mechanics problem, demonstrate the efficacy of the proposed approach. In particular, the score-based diffusion model, coupled with the physics-informed CNN surrogate, effectively learns geometrical features from provided prior samples, yielding better inverse estimation results compared to the state-of-the-art techniques.