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Physics-Informed Deep Inverse Operator Networks for Solving PDE Inverse Problems
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Proposes PI-DIONs, a physics-informed DeepONet-style architecture that learns PDE inverse-problem solution operators without labeled data, with stability-estimate-based generalization theory.
Problem
Solving PDE inverse problems (inverse source and permeability identification) with operator learning typically requires large labeled datasets of input-output pairs, and supervised models may ignore physics.
Approach
Parameterize both the solution u and target s as functions of coordinates via separate branch networks (reconstruction and inverse) sharing a trunk network, and train with physics residual, boundary, and partial-measurement data losses, so no labeled target is needed; theoretically connects these losses to L2 stability estimates for inverse problems.
Key results
- Unsupervised PI-DION achieves 1.03% relative L2 error on reaction-diffusion, beating supervised DeepONet (1.10% with 1000 samples) and FNO-based baselines.
- Supervised PI-DION achieves 5.64% on Helmholtz vs. 25.85% for FNO with 1000 samples; unsupervised PI-DION gets 8.05%.
- On Darcy flow, unsupervised PI-DION gets 8.10% vs. 3.41% for supervised FNO, so the unsupervised variant lags there.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use PI-DION instead of PI-DIONs-v0 architecture (ablation) | Reaction-Diffusion inverse source problem, supervised training · Reaction-Diffusion | relative L2 test error: 4.10% -> 1.04% | Helped |
| use PI-DION instead of PI-DIONs-v0 architecture (ablation) | Reaction-Diffusion inverse source problem, unsupervised training · Reaction-Diffusion | relative L2 test error: 5.62% -> 1.03% | Helped |
| increase λ2 from 1 to 100 while keeping λ1=1 (ablation) | PI-DION · Reaction-Diffusion | relative L2 error: 10.17% -> 1.03% | Helped |
| increase training samples from 100 to 500 (ablation) | PI-DION · Reaction-Diffusion | relative L2 error: 28.73% -> 5.07% | Helped |
| increase training samples from 500 to 1000 (ablation) | PI-DION · Reaction-Diffusion | relative L2 error: 5.07% -> 1.03% | Helped |
| increase training samples from 1000 to 2000 (ablation) | PI-DION · Reaction-Diffusion | relative L2 error: 1.03% -> 0.98% | Helped |
| use PI-DION instead of DeepONet | unsupervised PI-DION on the Reaction-Diffusion inverse source problem · Reaction-Diffusion | relative L2 error: 1.10% -> 1.03% | Helped |
| use supervised PI-DION instead of supervised FNO | Darcy Flow inverse permeability problem · Darcy Flow | relative L2 error: 3.41% -> 3.45% | Mixed |
| train PI-DION without target-function labels rather than with labels | PI-DION · Helmholtz | relative L2 error: 5.64% -> 8.05% | Hurt |
| train PI-DION without target-function labels rather than with labels | PI-DION · Reaction-Diffusion | relative L2 error: 1.04% -> 1.03% | No clear effect |
Surprising findings
- Unsupervised PI-DION outperforms supervised DeepONet on reaction-diffusion inverse source problem.
- Larger data-loss weight relative to physics-loss weight (lambda2/lambda1) consistently gave lower test error.
Limitations
- Unsupervised training is computationally expensive (5-24 h).
- Darcy flow unsupervised result (8.10%) is worse than supervised FNO (3.41%); no stability-based physics loss identified for that problem.
- Theoretical bounds on N, K, M, L not empirically verified.
Experimental setup
Datasets: Gaussian random field samples solved with finite difference method (reaction-diffusion, Helmholtz, Darcy flow)
Benchmarks: reaction-diffusion inverse source problem Helmholtz inverse source problem Darcy flow permeability inversion
Metrics: relative L2 error
Baselines: DeepONet (CNN/MLP branch) DeepONet (FNO branch) FNO PINN
Builds on: DeepONet PI-DeepONet Fourier Neural Operator Neural Inverse Operators physics-informed neural networks
Closest papers by meaning
- Latent Neural Operator for Solving Forward and Inverse PDE Problems (NeurIPS 2024 · similarity 0.89)
- Learning a Neural Solver for Parametric PDEs to Enhance Physics-Informed Methods (ICLR 2025 · similarity 0.88)
- RIGNO: A Graph-based Framework For Robust And Accurate Operator Learning For PDEs On Arbitrary Domains (NeurIPS 2025 · similarity 0.88)
- Data-Efficient Operator Learning via Unsupervised Pretraining and In-Context Learning (NeurIPS 2024 · similarity 0.88)
- SBNO : Schrödinger Bridge Neural Operator for the Forward and Inverse Problems under Missing Information (NeurIPS 2026 · similarity 0.87)
- PIED: Physics-Informed Experimental Design for Inverse Problems (ICLR 2025 · similarity 0.87)
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