Physics-Informed Deep Inverse Operator Networks for Solving PDE Inverse Problems

TL;DR

This work tackles PDE-based inverse problems by learning inverse operators without labeled data. The authors introduce Physics-Informed Deep Inverse Operator Networks (PI-DIONs), which jointly learn forward and inverse mappings as functions of space using a trunk-branch architecture and physics-informed losses that enforce PDE constraints and boundary conditions. They extend stability estimates from classical inverse problems to the operator-learning setting and prove a universal approximation theorem for PI-DIONs, ensuring convergence of the learned operators with enough data and capacity. Empirically, PI-DIONs demonstrate competitive performance against supervised baselines across reaction–diffusion, Helmholtz, and Darcy flow inverse problems, while offering real-time inference and data-efficiency advantages. The results suggest that integrating physics-based constraints into operator learning enables accurate, labeled-data-free solutions to challenging PDE inverse problems with broad practical impact.

Abstract

Inverse problems involving partial differential equations (PDEs) can be seen as discovering a mapping from measurement data to unknown quantities, often framed within an operator learning approach. However, existing methods typically rely on large amounts of labeled training data, which is impractical for most real-world applications. Moreover, these supervised models may fail to capture the underlying physical principles accurately. To address these limitations, we propose a novel architecture called Physics-Informed Deep Inverse Operator Networks (PI-DIONs), which can learn the solution operator of PDE-based inverse problems without labeled training data. We extend the stability estimates established in the inverse problem literature to the operator learning framework, thereby providing a robust theoretical foundation for our method. These estimates guarantee that the proposed model, trained on a finite sample and grid, generalizes effectively across the entire domain and function space. Extensive experiments are conducted to demonstrate that PI-DIONs can effectively and accurately learn the solution operators of the inverse problems without the need for labeled data.

Figures (6)

• Figure 1: Schematic illustration of PI-DIONs architecture. The reconstruction and inverse branch networks take partial measurement data as inputs and produce the coefficient vector for the solution and the target function, respectively. The trunk network takes the collocation point $x$ as input and generates the corresponding basis functions for both the solution and the target function. All the networks are trained by simultaneously minimizing the loss function $\mathcal{L}=\mathcal{L}_{physics}+\mathcal{L}_{data}$.
• Figure 2: A test sample and results for the inverse source problem of the reaction-diffusion equation. (a) True solution $u$ on a $30\times30$ rectangular grid. (b) Partial measurement data $u\vert_{\partial\Omega_T}$ collected from the true solution on the same grid. (c) Predicted solution $u_{\eta, \theta}$ on a $200\times200$ rectangular grid. (d) The source function $f(x)$ (blue) and the predicted source function $s_{\zeta, \theta}$ (red).
• Figure 3: A test sample and results for the Helmholtz equation. (a) Partial measurement $u\vert_{\Omega_m}$ on an internal $40\times40$ grid. (b) Predicted solution $u_{\eta, \theta}(x,y)$ on a $200\times200$ rectangular grid. (c) True source function $f(x)$ on a $50\times50$ grid. (d) Predicted source function $s_{\zeta, \theta}(x,y)$ on a $200\times200$ grid.
• Figure 4: A test sample and results for the Darcy flow. (a) Full measurement $u$ on a $30\times30$ rectangular grid. (b) Predicted solution $u_{\eta, \theta}$ on a $200\times200$ rectangular grid. (c) True permeability field $s$ on a $30\times30$ grid. (d) Predicted permeability field $s_{\zeta, \theta}$ on a $200\times200$.
• Figure 5: Schematic illustration of PI-DIONs-v0 architecture. The inverse branch network takes partial measurement data as input and produces the coefficient vector for the target function. The trunk network receives the collocation point $x$ as input and generates the corresponding basis functions for the target function. The forward branch network takes the predicted target function $s_{\zeta, \theta}$ as input and outputs the coefficients that form the solution $u_{\eta,\zeta, \theta}$. All the networks are trained by simultaneously minimizing the loss function $\mathcal{L}=\mathcal{L}_{physics}+\mathcal{L}_{data}$.

Theorems & Definitions (23)

• Theorem 1
• Theorem 2
• Theorem 3
• Remark 1
• Proposition 1
• Lemma 1
• proof
• Definition 1
• Lemma 2
• proof