Toward the Fully Physics-Informed Echo State Network -- an ODE Approximator Based on Recurrent Artificial Neurons

Echo State Networks trained by Tikhonov least squares are L2(μ) approximators of ergodic dynamical systems

Echo State Networks (ESNs) are a class of single-layer recurrent neural networks with randomly generated internal weights, and a single layer of tuneable outer weights, which are usually trained by regularised linear least squares regression. Remarkably, ESNs still enjoy the universal approximation property despite the training procedure being entirely linear. In this paper, we prove that an ESN trained on a sequence of observations from an ergodic dynamical system (with invariant measure $μ$) using Tikhonov least squares regression against a set of targets, will approximate the target functio…

用于可控动力系统建模的物理信息回声状态网络

Physics-Informed Echo State Networks for Modeling Controllable Dynamical Systems

本文将物理信息回声状态网络扩展至带外部输入的可控非线性系统，结合自适应平衡损失，在数据稀缺时显著降低过拟合和测试误差，并提升模型预测控制性能。

Unsupervised Reservoir Computing for Solving Ordinary Differential Equations

There is a wave of interest in using unsupervised neural networks for solving differential equations. The existing methods are based on feed-forward networks, {while} recurrent neural network differential equation solvers have not yet been reported. We introduce an unsupervised reservoir computing (RC), an echo-state recurrent neural network capable of discovering approximate solutions that satisfy ordinary differential equations (ODEs). We suggest an approach to calculate time derivatives of recurrent neural network outputs without using backpropagation. The internal weights of an RC are fixe…

Neural Abstractions

We present a novel method for the safety verification of nonlinear dynamical models that uses neural networks to represent abstractions of their dynamics. Neural networks have extensively been used before as approximators; in this work, we make a step further and use them for the first time as abstractions. For a given dynamical model, our method synthesises a neural network that overapproximates its dynamics by ensuring an arbitrarily tight, formally certified bound on the approximation error. For this purpose, we employ a counterexample-guided inductive synthesis procedure. We show that this…

可解释的多项式神经常微分方程

Interpretable Polynomial Neural Ordinary Differential Equations

针对标准 neural ODE 不可解释且训练区域外泛化差的问题，提出在其框架内嵌深度多项式神经网络的多项式神经 ODE，实现训练区域外预测与无需 SINDy 的直接符号回归。

Deep Galerkin 与 PINNs 方法求解偏微分方程的全局收敛性

Global Convergence of Deep Galerkin and PINNs Methods for Solving Partial Differential Equations

本文证明 Deep Galerkin 方法和 PINNs 在宽网络极限下全局收敛于偏微分方程的解，通过分析极限 ODE 中的核函数克服谱间隙挑战。