Self-adaptive weights based on balanced residual decay rate for physics-informed neural networks and deep operator networks

• URL: http://arxiv.org/abs/2407.01613v1
• Date: Fri, 28 Jun 2024 00:53:48 GMT
• Title: Self-adaptive weights based on balanced residual decay rate for physics-informed neural networks and deep operator networks
• Authors: Wenqian Chen, Amanda A. Howard, Panos Stinis,
• Abstract summary: Physics-informed deep learning has emerged as a promising alternative for solving partial differential equations. For complex problems, training these networks can still be challenging, often resulting in unsatisfactory accuracy and efficiency. We propose a point-wise adaptive weighting method that balances the residual decay rate across different training points.
• Score: 1.0562108865927007
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Physics-informed deep learning has emerged as a promising alternative for solving partial differential equations. However, for complex problems, training these networks can still be challenging, often resulting in unsatisfactory accuracy and efficiency. In this work, we demonstrate that the failure of plain physics-informed neural networks arises from the significant discrepancy in the convergence speed of residuals at different training points, where the slowest convergence speed dominates the overall solution convergence. Based on these observations, we propose a point-wise adaptive weighting method that balances the residual decay rate across different training points. The performance of our proposed adaptive weighting method is compared with current state-of-the-art adaptive weighting methods on benchmark problems for both physics-informed neural networks and physics-informed deep operator networks. Through extensive numerical results we demonstrate that our proposed approach of balanced residual decay rates offers several advantages, including bounded weights, high prediction accuracy, fast convergence speed, low training uncertainty, low computational cost and ease of hyperparameter tuning.

Related papers

• PhysicsCorrect: A Training-Free Approach for Stable Neural PDE Simulations [4.7903561901859355]
  We present PhysicsCorrect, a training-free correction framework that enforces PDE consistency at each prediction step.<n>Our key innovation is an efficient caching strategy that precomputes the Jacobian and its pseudoinverse during an offline warm-up phase.<n>Across three representative PDE systems, PhysicsCorrect reduces prediction errors by up to 100x while adding negligible inference time.
  arXiv  Detail & Related papers  (2025-07-03T01:22:57Z)
• The Butterfly Effect: Neural Network Training Trajectories Are Highly Sensitive to Initial Conditions [51.68215326304272]
  We show that even small perturbations reliably cause otherwise identical training trajectories to diverge-an effect that diminishes rapidly over training time.<n>Our findings provide insights into neural network training stability, with practical implications for fine-tuning, model merging, and diversity of model ensembles.
  arXiv  Detail & Related papers  (2025-06-16T08:35:16Z)
• Enriched Physics-informed Neural Networks for Dynamic Poisson-Nernst-Planck Systems [0.8192907805418583]
  This paper proposes a meshless deep learning algorithm, enriched physics-informed neural networks (EPINNs) to solve dynamic Poisson-Nernst-Planck (PNP) equations. The EPINNs takes the traditional physics-informed neural networks as the foundation framework, and adds the adaptive loss weight to balance the loss functions. Numerical results indicate that the new method has better applicability than traditional numerical methods in solving such coupled nonlinear systems.
  arXiv  Detail & Related papers  (2024-02-01T02:57:07Z)
• Implicit Stochastic Gradient Descent for Training Physics-informed Neural Networks [51.92362217307946]
  Physics-informed neural networks (PINNs) have effectively been demonstrated in solving forward and inverse differential equation problems. PINNs are trapped in training failures when the target functions to be approximated exhibit high-frequency or multi-scale features. In this paper, we propose to employ implicit gradient descent (ISGD) method to train PINNs for improving the stability of training process.
  arXiv  Detail & Related papers  (2023-03-03T08:17:47Z)
• NeuralStagger: Accelerating Physics-constrained Neural PDE Solver with Spatial-temporal Decomposition [67.46012350241969]
  This paper proposes a general acceleration methodology called NeuralStagger. It decomposing the original learning tasks into several coarser-resolution subtasks. We demonstrate the successful application of NeuralStagger on 2D and 3D fluid dynamics simulations.
  arXiv  Detail & Related papers  (2023-02-20T19:36:52Z)
• Physics-informed neural networks for pathloss prediction [0.9208007322096533]
  It is shown that the solution to a proposed learning problem improves generalization and prediction quality with a small number of neural network layers and parameters. The physics-informed formulation allows training and prediction with a small amount of training data which makes it appealing for a wide range of practical pathloss prediction scenarios.
  arXiv  Detail & Related papers  (2022-11-23T14:45:48Z)
• Tunable Complexity Benchmarks for Evaluating Physics-Informed Neural Networks on Coupled Ordinary Differential Equations [64.78260098263489]
  In this work, we assess the ability of physics-informed neural networks (PINNs) to solve increasingly-complex coupled ordinary differential equations (ODEs) We show that PINNs eventually fail to produce correct solutions to these benchmarks as their complexity increases. We identify several reasons why this may be the case, including insufficient network capacity, poor conditioning of the ODEs, and high local curvature, as measured by the Laplacian of the PINN loss.
  arXiv  Detail & Related papers  (2022-10-14T15:01:32Z)
• Green, Quantized Federated Learning over Wireless Networks: An Energy-Efficient Design [68.86220939532373]
  The finite precision level is captured through the use of quantized neural networks (QNNs) that quantize weights and activations in fixed-precision format. The proposed FL framework can reduce energy consumption until convergence by up to 70% compared to a baseline FL algorithm.
  arXiv  Detail & Related papers  (2022-07-19T16:37:24Z)
• Hierarchical Learning to Solve Partial Differential Equations Using Physics-Informed Neural Networks [2.0305676256390934]
  We propose a hierarchical approach to improve the convergence rate and accuracy of the neural network solution to partial differential equations. We validate the efficiency and robustness of the proposed hierarchical approach through a suite of linear and nonlinear partial differential equations.
  arXiv  Detail & Related papers  (2021-12-02T13:53:42Z)
• Efficient training of physics-informed neural networks via importance sampling [2.9005223064604078]
  Physics-In Neural Networks (PINNs) are a class of deep neural networks that are trained to compute systems governed by partial differential equations (PDEs) We show that an importance sampling approach will improve the convergence behavior of PINNs training.
  arXiv  Detail & Related papers  (2021-04-26T02:45:10Z)
• On Theory-training Neural Networks to Infer the Solution of Highly Coupled Differential Equations [0.0]
  We present insights into theory-training networks for learning the solution of highly coupled differential equations. We introduce a theory-training technique that, by leveraging regularization, eliminates those oscillations, decreases the final training loss, and improves the accuracy of the inferred solution.
  arXiv  Detail & Related papers  (2021-02-09T15:45:08Z)
• Adaptive Gradient Method with Resilience and Momentum [120.83046824742455]
  We propose an Adaptive Gradient Method with Resilience and Momentum (AdaRem) AdaRem adjusts the parameter-wise learning rate according to whether the direction of one parameter changes in the past is aligned with the direction of the current gradient. Our method outperforms previous adaptive learning rate-based algorithms in terms of the training speed and the test error.
  arXiv  Detail & Related papers  (2020-10-21T14:49:00Z)
• Revisiting Initialization of Neural Networks [72.24615341588846]
  We propose a rigorous estimation of the global curvature of weights across layers by approximating and controlling the norm of their Hessian matrix. Our experiments on Word2Vec and the MNIST/CIFAR image classification tasks confirm that tracking the Hessian norm is a useful diagnostic tool.
  arXiv  Detail & Related papers  (2020-04-20T18:12:56Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.