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Adversarial Adaptive Sampling: Unify PINN and Optimal Transport for the Approximation of PDEs
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Proposes adversarial adaptive sampling (AAS), a minmax scheme in which a KRnet normalizing flow adversarially reweights PINN collocation points to flatten the residual profile, reducing Monte Carlo variance.
Problem
Neural network (PINN) approximation of PDEs suffers statistical error from random collocation points, especially for low-regularity solutions, localized peaks, or high-dimensional problems, where uniform sampling is inefficient.
Approach
Formulate PINN training as a minmax problem where a normalizing-flow density p_alpha is maximized against the weighted squared residual (regularized by an H1 penalty, linked to a Wasserstein distance to uniform) while the solution network minimizes it, so the training set is adaptively concentrated and the residual profile is smoothed to reduce Monte Carlo variance.
Key results
- On the one-peak problem AAS reaches MSE 2.97e-05 versus 3.75e-04 for DAS-G and 1.93e-04 for DAS-R (table).
- On the two-peak problem AAS attains 1.09e-04 versus 1.51e-03 for DAS-G and 6.21e-03 for DAS-R.
- On the 10-d nonlinear problem AAS reaches 1.31e-03 versus 9.55e-03 for DAS-G and 1.26e-02 for DAS-R.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use AAS instead of PINN | PDE approximation · One-peak PDE | error: 9.74e-04 -> 2.97e-05 | Helped |
| Use AAS instead of DAS-R | PDE approximation · One-peak PDE | error: 1.93e-04 -> 2.97e-05 | Helped |
| Use AAS instead of PINN | PDE approximation · Two-peak PDE | error: 3.22e-02 -> 1.09e-04 | Helped |
| Use AAS instead of DAS-G | PDE approximation · Two-peak PDE | error: 1.51e-03 -> 1.09e-04 | Helped |
| Use AAS instead of PINN | PDE approximation · 10D nonlinear PDE | error: 1.01 -> 1.31e-03 | Helped |
| Use AAS instead of DAS-G | PDE approximation · 10D nonlinear PDE | error: 9.55e-03 -> 1.31e-03 | Helped |
| Vary the regularization parameter among the reported values (ablation) | AAS · One-peak PDE | error | No clear effect |
Limitations
- Theory relies on an assumption (surjectivity of residual operator) and a Lipschitz/H1 regularization that is only loosely connected to the implemented constraint.
- Only a few low-dimensional benchmark PDEs evaluated; code promised but not yet released.
Experimental setup
Datasets: synthetic PDE problems (one-peak, two-peak, 10-d nonlinear, parametric Burgers)
Benchmarks: one-peak Poisson problem two-peak problem 10-dimensional nonlinear PDE parametric Burgers' equation
Metrics: mean squared error on grid residual variance
Baselines: PINN RAR DAS-G DAS-R
Builds on: PINN WGAN deep adaptive sampling (DAS) KRnet
Closest papers by meaning
- PDE+: Enhancing Generalization via PDE with Adaptive Distributional Diffusion (AAAI 2024 · similarity 0.88)
- PINN Balls: Scaling Second-Order Methods for PINNs with Domain Decomposition and Adaptive Sampling (NeurIPS 2025 · similarity 0.88)
- Compute-optimal data scaling for neural surrogates via multi-fidelity training (NeurIPS 2026 · similarity 0.87)
- Learning to Discretize Denoising Diffusion ODEs (ICLR 2025 · similarity 0.87)
- Global convergence of adjoint-optimized neural PDEs (NeurIPS 2026 · similarity 0.86)
- CALM-PDE: Continuous and Adaptive Convolutions for Latent Space Modeling of Time-dependent PDEs (NeurIPS 2025 · similarity 0.86)
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