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PIED: Physics-Informed Experimental Design for Inverse Problems
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Proposes PIED, a fully differentiable experimental design framework using PINNs as forward simulators and inverse solvers, with Reptile meta-learned initialization and two criteria (FIST and MoTE) for one-shot sensor placement in inverse PDE problems.
Problem
One-shot experimental design (choosing observation inputs such as sensor placements) for inverse problems governed by PDEs, where existing ED methods need sequential adjustments and expensive numerical simulations.
Approach
Use PINNs as differentiable, meshless forward simulators and inverse solvers so that the design parameters can be optimized continuously by gradient descent; accelerate with Reptile meta-learned shared PINN initialization and approximate inverse-solver performance via few-step training (FIST) or empirical NTK kernel regression (MoTE).
Key results
- On the 2D Navier-Stokes task, MoTE achieves 1.18 (x10^-2) expected loss vs 4.33 for VBOED and 4.51 for Grid.
- On the function-valued Eikonal task, FIST and MoTE reach 0.74 and 0.76 (x10^1) vs 1.56 for Grid and 2.02 for MI.
- On the real groundwater dataset, FIST achieves 1.93 (x10^1) vs 2.10 for MI and 2.27 for Grid; on cell growth FIST reaches 2.62 vs 2.82 for VBOED.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use MoTE instead of the best baseline, MI | PIED observation selection · Wave (× 10⁻¹) | expected loss: 4.46 (0.92) -> 3.81 (2.34) | Helped |
| Use MoTE instead of the best baseline, VBOED | PIED observation selection · Navier-Stokes (× 10⁻²) | expected loss: 4.33 (0.98) -> 1.18 (0.11) | Helped |
| Use FIST instead of the best baseline, Grid | PIED observation selection · Eikonal (× 10¹) | expected loss: 1.56 (0.22) -> 0.74 (0.02) | Helped |
| Use FIST instead of the best baseline, MI | PIED observation selection · Groundwater (× 10¹) | expected loss: 2.10 (0.32) -> 1.93 (0.08) | Helped |
| Use FIST instead of the best baseline, VBOED | PIED observation selection · Cell Growth (× 10⁰) | expected loss: 2.82 (1.77) -> 2.62 (0.11) | Helped |
Surprising findings
- Benchmarks using classical numerical solvers with Bayesian optimization perform significantly worse than when the same benchmarks use PINNs as forward simulators.
Limitations
- Linearized (NTK) regime assumption for MoTE is unlikely to hold in practice for finite-width PINNs.
- Backpropagating through many gradient-descent steps in FIST becomes costly or non-differentiable for large step counts r.
Experimental setup
Datasets: simulated wave equation data simulated Navier-Stokes data (ANSYS Fluent) simulated Eikonal data (PyKonal) groundwater flow data scratch assay cell growth data
Benchmarks: Wave equation Navier-Stokes Eikonal Groundwater flow Cell growth
Metrics: median expected parameter estimation loss semi-interquartile range
Baselines: Random Grid Mutual information (MI) VBOED Numerical-solver-based ED with Bayesian optimization
Builds on: PINN Reptile neural tangent kernel
Closest papers by meaning
- Parameterized Physics-informed Neural Networks for Parameterized PDEs (ICML 2024 · similarity 0.88)
- Sub-Sequential Physics-Informed Learning with State Space Model (ICML 2025 · similarity 0.87)
- Physics-Informed Deep Inverse Operator Networks for Solving PDE Inverse Problems (ICLR 2025 · similarity 0.87)
- Gradient Alignment in Physics-informed Neural Networks: A Second-Order Optimization Perspective (NeurIPS 2025 · similarity 0.87)
- PINNacle: A Comprehensive Benchmark of Physics-Informed Neural Networks for Solving PDEs (NeurIPS 2024 · similarity 0.87)
- RoPINN: Region Optimized Physics-Informed Neural Networks (NeurIPS 2024 · similarity 0.86)
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