Source-linked AI summary
Physics-informed neural networks by Gradient-Guided Gaussian Adaptive Sampling (3GAS-PINNs)
Yousen Wang, Wei Zhao
TL;DR
Nonlinear PINNs face sparse sampling in complex or intermittent regions and difficulty capturing high-frequency physical features. The paper uses automatic differentiation within PINNs and reports substantially reduced errors for Burgers shocks and KdV colliding solitons, while noting sampling-related limitations.
Problem
Nonlinear PINNs can suffer from sparse sampling in complex regions and difficulty capturing high-frequency physical features.
Method
PINNs use Automatic Differentiation to compute exact partial derivatives of network outputs with respect to input coordinates.
Results
Across Burgers shocks and KdV colliding solitons, 3GAS-PINNs substantially reduces solution and gradient errors relative to conventional PINNs.
Takeaways & Limitations
Under equal or sparse collocation budgets, the reported error reductions indicate improved accuracy for these nonlinear PDE benchmarks.
Takeaways & Limitations
The reported sampling approach can produce sparse distributions in smooth regimes and low accuracy there.
Abstract
from arXiv · showhide
Physics-informed neural networks (PINNs) provide a mesh-free framework for solving partial differential equations, yet their performance in nonlinear problems is often limited by slow convergence, gradient imbalance, and insufficient resolution to capture localized intermittent structures such as shock waves[1]. These issues arise primarily from the use of fixed weights of loss and uniform collocation point distributions, which cannot adapt to the evolving complexity of the solution field during training. To address these challenges, Gradient-Guided Gaussian Adaptive Sampling Physics-Informed Neural Networks (3GAS-PINNs) is proposed in this paper, which combines uniform probability distribution and Gaussian-smoothed probability distribution derived from the spatial gradients of solution, to maintain global constraint satisfaction as well as concentrating collocation points in regions of high gradient. Thus, intermittency structures like shock wave and solitons can be accurately captured. The method is evaluated on three benchmark nonlinear problems, including one-dimensional forced Burgers equation, Korteweg-de Vries (KdV) equation and nonlinear Schrodinger equation, all of which exhibit steep gradients or strong nonlinearity. In comparison with baseline PINNs, 3GAS-PINNs can effectively promote the physical consistency in intermittent regions. The accuracy of the numerical simulation can be improved by a factor of up to 14.
1. Introduction
The introduction presents PINNs as a mesh-free, physics-constrained alternative for nonlinear PDE simulation, while emphasizing persistent challenges in resolving high-gradient intermittent structures. It motivates 3GAS-PINNs as a gradient-guided adaptive-sampling framework validated on three nonlinear equations.
- 1. Introduction: Traditional grid-based methods require excessively fine meshes to capture high-gradient structures in nonlinear systems, increasing computational cost sharply.The introduction highlights this issue particularly for high-Reynolds-number turbulence simulations.
- 1. Introduction: PINNs incorporate governing PDEs into the loss function, use automatic differentiation for derivatives, and combine residuals with boundary and initial-condition constraints.This provides a mesh-free route for approximating PDE solutions without relying solely on labeled data.
- 1. Introduction: Standard PINNs struggle with strong nonlinearities because fixed loss weights can conflict between boundary, initial, and PDE constraints.This imbalance can cause boundary fitting to take priority over internal dynamics and reduce physical consistency.
- 1. Introduction: Uniform or random collocation sampling oversamples smooth regions and undersamples critical areas, smearing shock fronts and reducing accuracy.The resulting allocation does not adapt computational resources to local physical complexity.
- 1. Introduction: Prior adaptive-sampling approaches improve collocation allocation and convergence efficiency but are described as focusing on sampling and loss deviations rather than the gradient imbalance itself.The paper positions 3GAS-PINNs as addressing high-frequency physical features and intermittent-region consistency through gradient-guided sampling.
- 1. Introduction: 3GAS-PINNs combines gradient-guided Gaussian probability sampling with PINNs to improve convergence, precision, and physical consistency in high-gradient intermittent regions.The method is evaluated on Burgers, KdV, and nonlinear Schrödinger equations.
2. Gradient-Guided Gaussian Adaptive Sampling PINNs (3GAS-PINNs)
3GAS-PINNs augments standard PINNs with gradient-guided adaptive sampling to better resolve localized high-gradient structures while retaining global physical constraints.
- 2. Gradient-Guided Gaussian Adaptive Sampling PINNs (3GAS-PINNs): Standard PINNs use automatic differentiation for exact input derivatives and optimize a composite loss covering PDE residual, initial-condition, and boundary-condition violations.The loss components are evaluated over discrete points, commonly using mean squared error.
- 2. Gradient-Guided Gaussian Adaptive Sampling PINNs (3GAS-PINNs): 3GAS-PINNs dynamically adjusts the objective function and collocation-point distribution to better capture localized intermittent structures.The method is designed to address failures of uniform sampling in regions with steep gradients.
- 2.2.1 Deficiency of uniformly random sampling in PINNs: Uniform sampling under-resolves intermittent regions with steep gradients, producing substantial deviations from reference solutions such as FDM results.The Burgers-equation example attributes this problem to insufficient sampling points near shock-wave fronts.
- 2.2.2 Gradient-Guided Gaussian Adaptive Sampling: The method computes gradient magnitudes, smooths their spatial distribution into an importance manifold, and normalizes it into an adaptive sampling density.The smoothed field reflects both local gradient magnitude and its relative significance across the global solution field.
- 2.2.2 Gradient-Guided Gaussian Adaptive Sampling: The Gaussian kernel width controls spatial extensibility: overly wide kernels make sampling too uniform, whereas overly narrow kernels create isolated clusters.The adaptive width is based on gradient and solution statistics and includes an empirical constant.
- 2.2.2 Gradient-Guided Gaussian Adaptive Sampling: Hybrid sampling combines gradient-guided points with uniformly sampled points, concentrating resolution in intermittent regions while preserving global PDE constraints.The adaptive subset is sampled from the normalized density and the remaining points are sampled uniformly.
3. Numerical simulation results
Across Burgers, KdV, and nonlinear Schrödinger benchmarks, 3GAS-PINNs adaptively concentrates collocation points near high-gradient intermittent structures and generally improves accuracy over PINNs.
- Adaptive sampling: Adaptive sampling concentrates points toward intermittent regimes when η is sufficiently large, including the Burgers shock region and KdV soliton structures.The reported concentration thresholds are η≥0.7 for Burgers and η≥0.5 for KdV.
- Adaptive sampling: Over-adaptation can reduce performance in smooth regions, while excessive Gaussian smoothing can obscure intermittent structures and weaken accuracy.The Burgers results identify sparse smooth-region sampling as a source of lower accuracy, and the KdV results report smearing when A increases.
- Burgers equation: For Burgers, 3GAS-PINNs reaches 10−3 loss in 4900 epochs, which is 31% of the epochs required by conventional PINNs.The method reaches 10−4 loss in 16819 epochs under the reported configuration.
- Burgers equation: For Burgers shocks, 3GAS-PINNs reduces solution error from 9.11 × 10−3 to 1.33 × 10−3 and gradient error from 8.27 × 10−2 to 7.13 × 10−2.These reductions correspond to approximately sevenfold and 16% accuracy improvements, respectively.
- Korteweg-de Vries equation: For KdV soliton interactions, 3GAS-PINNs reduces solution error from 6.11 × 10−3 to 1.83 × 10−3 and gradient error from 1.55 × 10−2 to 7.71 × 10−3.The reported improvements are approximately threefold for solution accuracy and twofold for gradient precision.
- Nonlinear Schrödinger equation: 3GAS-PINNs maintains lower solution and gradient errors across sampling budgets for the nonlinear Schrödinger equation, especially with few points.At 100 sampling points, the reported solution and gradient-error improvements are 14-fold and 11-fold relative to PINNs.
4. Discussion
3GAS-PINNs combines gradient-guided Gaussian smoothing with hybrid sampling to focus collocation on intermittent structures while preserving full-domain constraints. Across nonlinear PDEs, its problem-specific Gaussian scaling and targeted redistribution improve accuracy under equal or sparse computational budgets.
- Sampling mechanism: Dynamic redistribution accelerates gradient propagation across steep interfaces while reconstructing large-scale background flow and microscale intermittent structures with full-domain physical consistency.This targeted strategy avoids global domain searching and delivers higher sampling efficiency than the described metaheuristic approach.
- Parameter effects: The optimal Gaussian scaling coefficient differs across Burgers, KdV, and nonlinear Schrödinger equations because their intermittent structures have different characteristic spatial scales.The coefficient controls the spatial scope of intensified sampling near high-gradient zones.
- Parameter effects: For Burgers, A=1 balances shock-front and smooth-field coverage, whereas smaller or larger values respectively overconcentrate or blur refinement.For KdV, A=0.95 covers soliton collision bands; for NLSE, A=0.1 centralizes points around the rogue-wave gradient spike.
- Comparison with prior sampling: Residual-based sampling can misallocate points because early numerical noise creates false high residuals in smooth regions and delays refinement of moving intermittent structures.Gradient-guided sampling instead uses the current solution’s real-time gradient field as a direct local-structure indicator.
- Sampling mechanism: Gradient-guided sampling uses automatic-differentiation gradients, Gaussian smoothing, and hybrid uniform refinement to target high-gradient regions without abandoning global PDE constraints.The resulting redistribution responds synchronously to traveling intermittent features and avoids indirect residual-feedback lag.
- Overall accuracy: 3GAS-PINNs reduces Burgers shock error to 14.6% of conventional PINNs and reduces KdV solution and gradient errors by 3 and 2 folds, respectively.For NLSE rogue waves, its solution and gradient errors are 1/14 and 1/11 of those from conventional PINNs under sparse collocation.
5. Conclusion
The conclusion presents adaptive sampling and weighting as mechanisms for concentrating collocation on large-variation regions while mitigating gradient hysteresis. Experiments on KdV and nonlinear Schrödinger equations report accuracy gains of 2–14 times over traditional PINNs with fewer computational resources.
- Conclusion: The proposed adaptive sampling successfully captures complex physical features while obtaining accurate solutions with significantly fewer computational resources.This conclusion is supported by the reported nonlinear Schrödinger results.
- Conclusion: 3GAS-PINNs concentrates collocation points where the current solution varies strongly and uses adaptive loss weighting to balance different physical constraints.The stated mechanism is intended to mitigate gradient hysteresis in PINNs.