Convergence Analysis of Natural Gradient Descent for Over-parameterized Physics-Informed Neural Networks

• URL: http://arxiv.org/abs/2408.00573v4
• Date: Fri, 13 Jun 2025 11:46:16 GMT
• Title: Convergence Analysis of Natural Gradient Descent for Over-parameterized Physics-Informed Neural Networks
• Authors: Xianliang Xu, Ting Du, Wang Kong, Bin Shan, Ye Li, Zhongyi Huang,
• Abstract summary: We show that for training two-layer $textReLU3$ Physics-Informed Neural Networks (PINNs), the learning rate can be improved from $mathcalO(lambda_0)$ to $mathcalO (1/|bmHinfty|_2)$.<n>Despite such improvements, the convergence rate is still tied to the least eigenvalue of the Gram matrix, leading to slow convergence.
• Score: 4.554284689395686
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: In the context of over-parameterization, there is a line of work demonstrating that randomly initialized (stochastic) gradient descent (GD) converges to a globally optimal solution at a linear convergence rate for the quadratic loss function. However, the learning rate of GD for training two-layer neural networks exhibits poor dependence on the sample size and the Gram matrix, leading to a slow training process. In this paper, we show that for training two-layer $\text{ReLU}^3$ Physics-Informed Neural Networks (PINNs), the learning rate can be improved from $\mathcal{O}(\lambda_0)$ to $\mathcal{O}(1/\|\bm{H}^{\infty}\|_2)$, implying that GD actually enjoys a faster convergence rate. Despite such improvements, the convergence rate is still tied to the least eigenvalue of the Gram matrix, leading to slow convergence. We then develop the positive definiteness of Gram matrices with general smooth activation functions and provide the convergence analysis of natural gradient descent (NGD) in training two-layer PINNs, demonstrating that the learning rate can be $\mathcal{O}(1)$ and at this rate, the convergence rate is independent of the Gram matrix. In particular, for smooth activation functions, the convergence rate of NGD is quadratic. Numerical experiments are conducted to verify our theoretical results.

Related papers

• Preconditioned Gradient Descent Finds Over-Parameterized Neural Networks with Sharp Generalization for Nonparametric Regression [8.130817534654089]
  We consider nonparametric regression by a two-layer neural network trained by gradient descent (GD) or its variant in this paper. We show that, if the neural network is trained with a novel Preconditioned Gradient Descent (PGD) with early stopping and the target function has spectral bias widely studied in the deep learning literature, the trained network renders a particularly sharp generalization bound with a minimax optimal rate of $cO(1/n4alpha/(4alpha+1)$.
  arXiv  Detail & Related papers  (2024-07-16T03:38:34Z)
• Convergence of Implicit Gradient Descent for Training Two-Layer Physics-Informed Neural Networks [3.680127959836384]
  implicit gradient descent (IGD) outperforms the common gradient descent (GD) in handling certain multi-scale problems. We show that IGD converges a globally optimal solution at a linear convergence rate.
  arXiv  Detail & Related papers  (2024-07-03T06:10:41Z)
• Learning with Norm Constrained, Over-parameterized, Two-layer Neural Networks [54.177130905659155]
  Recent studies show that a reproducing kernel Hilbert space (RKHS) is not a suitable space to model functions by neural networks. In this paper, we study a suitable function space for over- parameterized two-layer neural networks with bounded norms.
  arXiv  Detail & Related papers  (2024-04-29T15:04:07Z)
• A Mean-Field Analysis of Neural Stochastic Gradient Descent-Ascent for Functional Minimax Optimization [90.87444114491116]
  This paper studies minimax optimization problems defined over infinite-dimensional function classes of overparametricized two-layer neural networks. We address (i) the convergence of the gradient descent-ascent algorithm and (ii) the representation learning of the neural networks. Results show that the feature representation induced by the neural networks is allowed to deviate from the initial one by the magnitude of $O(alpha-1)$, measured in terms of the Wasserstein distance.
  arXiv  Detail & Related papers  (2024-04-18T16:46:08Z)
• Stable Nonconvex-Nonconcave Training via Linear Interpolation [51.668052890249726]
  This paper presents a theoretical analysis of linearahead as a principled method for stabilizing (large-scale) neural network training. We argue that instabilities in the optimization process are often caused by the nonmonotonicity of the loss landscape and show how linear can help by leveraging the theory of nonexpansive operators.
  arXiv  Detail & Related papers  (2023-10-20T12:45:12Z)
• Fast Convergence in Learning Two-Layer Neural Networks with Separable Data [37.908159361149835]
  We study normalized gradient descent on two-layer neural nets. We prove for exponentially-tailed losses that using normalized GD leads to linear rate of convergence of the training loss to the global optimum.
  arXiv  Detail & Related papers  (2023-05-22T20:30:10Z)
• Gauss-Newton Temporal Difference Learning with Nonlinear Function Approximation [11.925232472331494]
  A Gauss-Newton Temporal Difference (GNTD) learning method is proposed to solve the Q-learning problem with nonlinear function approximation. In each iteration, our method takes one Gauss-Newton (GN) step to optimize a variant of Mean-Squared Bellman Error (MSBE) We validate our method via extensive experiments in several RL benchmarks, where GNTD exhibits both higher rewards and faster convergence than TD-type methods.
  arXiv  Detail & Related papers  (2023-02-25T14:14:01Z)
• Over-Parameterization Exponentially Slows Down Gradient Descent for Learning a Single Neuron [49.45105570960104]
  We prove the global convergence of randomly gradient descent with a $Oleft(T-3right)$ rate. These two bounds jointly give an exact characterization of the convergence rate. We show this potential function converges slowly, which implies the slow convergence rate of the loss function.
  arXiv  Detail & Related papers  (2023-02-20T15:33:26Z)
• Implicit Bias in Leaky ReLU Networks Trained on High-Dimensional Data [63.34506218832164]
  In this work, we investigate the implicit bias of gradient flow and gradient descent in two-layer fully-connected neural networks with ReLU activations. For gradient flow, we leverage recent work on the implicit bias for homogeneous neural networks to show that leakyally, gradient flow produces a neural network with rank at most two. For gradient descent, provided the random variance is small enough, we show that a single step of gradient descent suffices to drastically reduce the rank of the network, and that the rank remains small throughout training.
  arXiv  Detail & Related papers  (2022-10-13T15:09:54Z)
• A Convergence Analysis of Nesterov's Accelerated Gradient Method in Training Deep Linear Neural Networks [21.994004684742812]
  Momentum methods are widely used in training networks for their fast trajectory. We show that the convergence of the random number and $kappaO can converge to the global minimum. We extend our analysis to deep linear ResNets and derive a similar result.
  arXiv  Detail & Related papers  (2022-04-18T13:24:12Z)
• Leveraging Non-uniformity in First-order Non-convex Optimization [93.6817946818977]
  Non-uniform refinement of objective functions leads to emphNon-uniform Smoothness (NS) and emphNon-uniform Lojasiewicz inequality (NL) New definitions inspire new geometry-aware first-order methods that converge to global optimality faster than the classical $Omega (1/t2)$ lower bounds.
  arXiv  Detail & Related papers  (2021-05-13T04:23:07Z)
• Regularization Matters: A Nonparametric Perspective on Overparametrized Neural Network [20.132432350255087]
  Overparametrized neural networks trained by tangent descent (GD) can provably overfit any training data. This paper studies how well overparametrized neural networks can recover the true target function in the presence of random noises.
  arXiv  Detail & Related papers  (2020-07-06T01:02:23Z)
• Provably Efficient Neural Estimation of Structural Equation Model: An Adversarial Approach [144.21892195917758]
  We study estimation in a class of generalized Structural equation models (SEMs) We formulate the linear operator equation as a min-max game, where both players are parameterized by neural networks (NNs), and learn the parameters of these neural networks using a gradient descent. For the first time we provide a tractable estimation procedure for SEMs based on NNs with provable convergence and without the need for sample splitting.
  arXiv  Detail & Related papers  (2020-07-02T17:55:47Z)
• Towards Better Understanding of Adaptive Gradient Algorithms in Generative Adversarial Nets [71.05306664267832]
  Adaptive algorithms perform gradient updates using the history of gradients and are ubiquitous in training deep neural networks. In this paper we analyze a variant of OptimisticOA algorithm for nonconcave minmax problems. Our experiments show that adaptive GAN non-adaptive gradient algorithms can be observed empirically.
  arXiv  Detail & Related papers  (2019-12-26T22:10:10Z)

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.