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

Yankun Hong, Harshit Bansal, Karen Veroy

TL;DR

This work tackles PDE-constrained inverse problems with highly high-dimensional parameters by introducing a score-based diffusion framework that learns priors from samples and performs posterior sampling via a refined noising-denoising process. A time-varying step-size strategy addresses instabilities in conditional diffusion, and a physics-informed CNN surrogate accelerates forward-model evaluations while preserving physical fidelity. The approach is validated on hyper-elastic and multi-scale mechanics problems, showing superior reconstruction of geometrical features from boundary observations compared to ensemble Kalman-based methods, especially when priors are complex. The combination of learned priors, posterior diffusion sampling, and physics-informed surrogates offers a scalable, uncertainty-aware solver for challenging PDE-constrained inverse problems with rich prior information.

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.

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

TL;DR

This work tackles PDE-constrained inverse problems with highly high-dimensional parameters by introducing a score-based diffusion framework that learns priors from samples and performs posterior sampling via a refined noising-denoising process. A time-varying step-size strategy addresses instabilities in conditional diffusion, and a physics-informed CNN surrogate accelerates forward-model evaluations while preserving physical fidelity. The approach is validated on hyper-elastic and multi-scale mechanics problems, showing superior reconstruction of geometrical features from boundary observations compared to ensemble Kalman-based methods, especially when priors are complex. The combination of learned priors, posterior diffusion sampling, and physics-informed surrogates offers a scalable, uncertainty-aware solver for challenging PDE-constrained inverse problems with rich prior information.

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.
Paper Structure (20 sections, 1 theorem, 47 equations, 14 figures, 3 algorithms)

This paper contains 20 sections, 1 theorem, 47 equations, 14 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Outline
  3. Problem description and Bayesian inversion
  4. Score-based generative diffusion model
  5. Diffusion model for unconditional distribution
  6. Noising-denoising process
  7. Denoising score matching
  8. Diffusion model for posterior distribution
  9. Standard posterior sampling
  10. Time-varying step size
  11. Forward model surrogate
  12. Model Problems
  13. Convolutional neural network
  14. Physics-informed strategy
  15. Results and discussion
  16. ...and 5 more sections

Key Result

Theorem 1

Let the conditional distribution $p(\boldsymbol{\mu}_{t_n}|\boldsymbol{\mu}_0)$ belong to the exponential family that has the following formulation: where $F$ is a function of $\boldsymbol{\mu}_{t_n}$, $\phi(\boldsymbol{\mu}_0)$ is a function that normalizes the density, and $p_0$ is the density when $\boldsymbol{\mu}_0 = 0$. Under the aforementioned setting, the posterior mean $\bar{\boldsymbol{

Figures (14)

  • Figure 1: The schematic diagram for the noising and denoising processes.
  • Figure 2: Computational domains and meshes.
  • Figure 3: The absolute errors of the CNN and physics-informed CNN surrogates (orange and green, respectively), and the standard deviation within the prior sample $S_0$ (blue) at each boundary observation point. The observation points along the $x$-axis are sorted in decreasing order of the standard deviation. The red lines represent different observation error settings.
  • Figure 4: 24 samples of prior
  • Figure 5: (1) the "true" mean of the given prior $S$; (2) the generated mean via SMLD; (3) the difference between (1) and (2); (4) the generated mean via DDPM; (5) the difference between (1) and (4).
  • ...and 9 more figures

Theorems & Definitions (3)

  • Remark
  • Theorem 1: Tweedie's formula efron_tweedies_2011
  • Remark