This paper analyzes the inverse relationship between the order of partial differential equations (PDEs) and the convergence of gradient descent in physics-informed neural networks (PINNs) with the power of ReLU activation. The integration of the PDE into a loss function endows PINNs with a distinctive feature to require computing derivatives of model up to the PDE order. Although it has been empirically observed that PINNs encounter difficulties in convergence when dealing with high-order or high-dimensional PDEs, a comprehensive theoretical understanding of this issue remains elusive. This paper offers theoretical support for this pathological behavior by demonstrating that the gradient flow converges in a lower probability when the PDE order is higher. In addition, we show that PINNs struggle to address high-dimensional problems because the influence of dimensionality on convergence is exacerbated with increasing PDE order. To address the pathology, we use the insights garnered to consider variable splitting that decomposes the high-order PDE into a system of lower-order PDEs. We prove that by reducing the differential order, the gradient flow of variable splitting is more likely to converge to the global optimum. Furthermore, we present numerical experiments in support of our theoretical claims.

A recurrent neural network (RNN) is a widely used deep-learning network for dealing with sequential data. Imitating a dynamical system, an infinite-width RNN can approximate any open dynamical system in a compact domain. In general, deep narrow networks with bounded width and arbitrary depth are more effective than wide shallow networks with arbitrary width and bounded depth in practice; however, the universal approximation theorem for deep narrow structures has yet to be extensively studied. In this study, we prove the universality of deep narrow RNNs and show that the upper bound of the minimum width for universality can be independent of the length of the data. Specifically, we show a deep RNN with ReLU activation can approximate any continuous function or $L^p$ function with the widths $d_x+d_y+3$ and $\max\{d_x+1,d_y\}$, respectively, where the target function maps a finite sequence of vectors in $\mathbb{R}^{d_x}$ to a finite sequence of vectors in $\mathbb{R}^{d_y}$. We also compute the additional width required if the activation function is sigmoid or more. In addition, we prove the universality of other recurrent networks, such as bidirectional RNNs. Bridging a multi-layer perceptron and an RNN, our theory and technique can shed light on further research on deep RNNs.