Efficient Global-Local Fusion Sampling for Physics-Informed Neural Networks

TL;DR

This work tackles the sensitivity of Physics-Informed Neural Networks (PINNs) to collocation-point placement, a key factor in accurately approximating PDE residuals. It introduces Global-Local Fusion (GLF) sampling, which combines residual-adaptive neighborhood sampling with a lightweight residual-distribution surrogate to retain global stability while enabling local refinement. The method comprises two main components: (i) Local Residual-driven Probabilistic Sampling that concentrates samples around high-residual regions using $h(\boldsymbol{x})=\frac{\alpha}{r(\boldsymbol{x})+\epsilon}$, and (ii) Global Distribution-Based Selection that uses an inexpensive approximation of the residual distribution to pick new points without expensive derivatives. Extensive benchmarks show that GLF consistently improves accuracy and reduces memory usage compared with global and purely local samplers, providing a scalable, stable framework for solving high-dimensional PDEs with PINNs.

Abstract

The accuracy of Physics-Informed Neural Networks (PINNs) critically depends on the placement of collocation points, as the PDE loss is approximated through sampling over the solution domain. Global sampling ensures stability by covering the entire domain but requires many samples and is computationally expensive, whereas local sampling improves efficiency by focusing on high-residual regions but may neglect well-learned areas, reducing robustness. We propose a Global-Local Fusion (GLF) Sampling Strategy that combines the strengths of both approaches. Specifically, new collocation points are generated by perturbing training points with Gaussian noise scaled inversely to the residual, thereby concentrating samples in difficult regions while preserving exploration. To further reduce computational overhead, a lightweight linear surrogate is introduced to approximate the global residual-based distribution, achieving similar effectiveness at a fraction of the cost. Together, these components, residual-adaptive sampling and residual-based approximation, preserve the stability of global methods while retaining the efficiency of local refinement. Extensive experiments on benchmark PDEs demonstrate that GLF consistently improves both accuracy and efficiency compared with global and local sampling strategies. This study provides a practical and scalable framework for enhancing the reliability and efficiency of PINNs in solving complex and high-dimensional PDEs.

Figures (8)

• Figure 1: Algorithm flow. (a) Compute the residuals at training points. (b) Generate $M$ candidate points around each training point (anchor). (c) Assign each candidate the residual value of its anchor point (colors indicate different residual values). (d) Apply the approximate distribution to sample $N$ new points to update the training data.
• Figure 2: Effect of a residual-dependent scaling factor $h(\mathbf{x})$ on the distribution. (a) Standard normal distribution. (b) Distribution with a large scaling factor $h(\mathbf{x})$, corresponding to a small residual $r(\mathbf{x})$, resulting in a wider spread. (c) Distribution with a small scaling factor $h(\mathbf{x})$, corresponding to a large residual $r(\mathbf{x})$, leading to tighter local concentration.
• Figure 3: Piecewise-linear approximation with different numbers of training points. (a) With sparse training points, the approximation is coarse and deviates from the original distribution. (b) With dense training points, the approximation becomes finer and closely aligns with the original curve.
• Figure 4: Training loss curves for different PDEs. Red line: Mean values of GLF method; Blue line: Mean values of RAD-100000 method; Green line: Mean values of RSmote method. The shaded areas represent the corresponding standard deviations.
• Figure 5: Solution fields for Allen-Cahn Equation. (a)-(c): RAD solution, RSmote solution and GLF solution; (d)-(f): Absolute differences; (g)-(i): The distribution of the final points.

Theorems & Definitions (1)

• Remark 1