A unified physics-informed generative operator framework for general inverse problems

TL;DR

This work introduces IGNO, a unified physics-informed generative neural operator framework for general PDE inverse problems. IGNO encodes high-dimensional coefficients and boundary conditions into a compact latent space using dual encoders and reconstructs coefficients and PDE solutions with two MultiONet-based decoders, all trained purely from PDE residuals without labeled input–output pairs. Inversion is performed by gradient-based optimization in latent space, aided by a normalizing flow that provides principled initializations. Across continuous and discontinuous coefficients and an operator-based EIT problem, IGNO demonstrates 3–6x accuracy improvements over state-of-the-art methods, robust performance under high noise, and strong generalization to out-of-distribution targets, enabling efficient, scalable, and versatile PDE inverse solving in diverse domains.

Abstract

Solving inverse problems governed by partial differential equations (PDEs) is central to science and engineering, yet remains challenging when measurements are sparse, noisy, or when the underlying coefficients are high-dimensional or discontinuous. Existing deep learning approaches either require extensive labeled datasets or are limited to specific measurement types, often leading to failure in such regimes and restricting their practical applicability. Here, a novel generative neural operator framework, IGNO, is introduced to overcome these limitations. IGNO unifies the solution of inverse problems from both point measurements and operator-valued data without labeled training pairs. This framework encodes high-dimensional, potentially discontinuous coefficient fields into a low-dimensional latent space, which drives neural operator decoders to reconstruct both coefficients and PDE solutions. Training relies purely on physics constraints through PDE residuals, while inversion proceeds via efficient gradient-based optimization in latent space, accelerated by an a priori normalizing flow model. Across a diverse set of challenging inverse problems, including recovery of discontinuous coefficients from solution-based measurements and the EIT problem with operator-based measurements, IGNO consistently achieves accurate, stable, and scalable inversion even under severe noise. It consistently outperforms the state-of-the-art method under varying noise levels and demonstrates strong generalization to out-of-distribution targets. These results establish IGNO as a unified and powerful framework for tackling challenging inverse problems across computational science domains.

Figures (8)

• Figure 1: The IGNO framework for PDE inverse problems. The generative neural operator architecture encodes the coefficient field $a$ and the boundary condition $g$ into a structured latent variable $\bm{\beta} = (\bm{\beta}_1, \bm{\beta}_2)$ via encoders $E_{\boldsymbol{\theta}_{\bm{\beta}_1}}$ and $E_{\boldsymbol{\theta}_{\bm{\beta}_2}}$, respectively. Two MultiONet-based decoders reconstruct the coefficient $a$ from $\bm{\beta}_1$ (coefficient decoder $\mathcal{G}_{\boldsymbol{\theta}_a}$) and predict the PDE solution $u$ from $\bm{\beta}$ (solution decoder $\mathcal{G}_{\boldsymbol{\theta}_u}$). The model is trained in a purely physics-informed manner, driven solely by governing PDE residuals without requiring paired input-output data. For inversion, gradient-based optimization in the low-dimensional latent space enables efficient recovery of unknown coefficients from sparse, noisy observations.
• Figure 1: The MultiONet architecture.
• Figure 2: Recovery of continuous coefficients with solution-based measurements (In-distribution case): (a) the ground truth permeability field $k$ and the corresponding pressure field $p$ (black dots denote $m=100$ random sensors); (b) recovered permeability $k^{rec}$ and corresponding pointwise absolute errors obtained by IGNO under different noise levels; (c) recovered permeability $k^{rec}$ and corresponding pointwise absolute errors obtained by PI-DIONs under different noise levels.
• Figure 2: Illustration of operator-based measurements in the EIT problem: (a) An example of conductivity $\gamma$ (left) and the boundary sensors $\bm{X}_m$ (right); (b) Examples of boundary conditions $g_l$ (left), corresponding current measurements $\Lambda_{\gamma}[g_l]$ (middle), and PDE solutions $u$ (right), when $l=1,10,20$.
• Figure 3: Recovery of piecewise-constant coefficients with solution-based measurements (In-distribution case): (a) the true permeability field $k$ (Phase 1 shown in yellow) and the corresponding pressure field $p$ (black dots denote $m=100$ random sensors); (b) recovered permeability $k^{rec}$ and corresponding pointwise absolute errors obtained by IGNO under different noise levels; (c) recovered permeability $k^{rec}$ and corresponding pointwise absolute errors obtained by PI-DIONs under different noise levels.