Ting Du

Xianliang Xu, Ting Du, Wang Kong et al.

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 $\mathcal{O}(λ_0)$ 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.

Ye Li, Ting Du, Yiwen Pang et al.

Solving Singularly Perturbed Differential Equations (SPDEs) poses computational challenges arising from the rapid transitions in their solutions within thin regions. The effectiveness of deep learning in addressing differential equations motivates us to employ these methods for solving SPDEs. In this manuscript, we introduce Component Fourier Neural Operator (ComFNO), an innovative operator learning method that builds upon Fourier Neural Operator (FNO), while simultaneously incorporating valuable prior knowledge obtained from asymptotic analysis. Our approach is not limited to FNO and can be applied to other neural network frameworks, such as Deep Operator Network (DeepONet), leading to potential similar SPDEs solvers. Experimental results across diverse classes of SPDEs demonstrate that ComFNO significantly improves accuracy compared to vanilla FNO. Furthermore, ComFNO exhibits natural adaptability to diverse data distributions and performs well in few-shot scenarios, showcasing its excellent generalization ability in practical situations.

Xianliang Xu, Ting Du, Wang Kong et al.

The optimization algorithms are crucial in training physics-informed neural networks (PINNs), as unsuitable methods may lead to poor solutions. Compared to the common gradient descent (GD) algorithm, implicit gradient descent (IGD) outperforms it in handling certain multi-scale problems. In this paper, we provide convergence analysis for the IGD in training over-parameterized two-layer PINNs. We first derive the training dynamics of IGD in training two-layer PINNs. Then, over-parameterization allows us to prove that the randomly initialized IGD converges to a globally optimal solution at a linear convergence rate. Moreover, due to the distinct training dynamics of IGD compared to GD, the learning rate can be selected independently of the sample size and the least eigenvalue of the Gram matrix. Additionally, the novel approach used in our convergence analysis imposes a milder requirement on the network width. Finally, empirical results validate our theoretical findings.