Neural networks are increasingly recognized as a powerful numerical solution technique for partial differential equations (PDEs) arising in diverse scientific computing domains, including quantum many-body physics. In the context of time-dependent PDEs, the dominant paradigm involves casting the approximate solution in terms of stochastic minimization of an objective function given by the norm of the PDE residual, viewed as a function of the neural network parameters. Recently, advancements have been made in the direction of an alternative approach which shares aspects of nonlinearly parametrized Galerkin methods and variational quantum Monte Carlo, especially for high-dimensional, time-dependent PDEs that extend beyond the usual scope of quantum physics. This paper is inspired by the potential of solving Hamilton-Jacobi-Bellman (HJB) PDEs using Neural Galerkin methods and commences the exploration of nonlinearly parametrized trial functions for which the evolution equations are analytically tractable. As a precursor to the Neural Galerkin scheme, we present trial functions with evolution equations that admit closed-form solutions, focusing on time-dependent HJB equations relevant to finance.

翻译：神经网络被越来越多地视为求解偏微分方程（PDEs）的强大数值技术，广泛应用于包括量子多体物理在内的多种科学计算领域。在时变PDEs的背景下，主流范式是通过将PDE残差范数作为神经网络参数的函数，对目标函数进行随机最小化来构建近似解。近年来，一种融合非线性参数化Galerkin方法与变分量子蒙特卡洛思想的替代方案取得了进展，尤其适用于超越量子物理常规范围的高维时变PDEs。本文受神经Galerkin方法求解哈密顿-雅可比-贝尔曼（HJB）PDEs潜力的启发，开始探索演化方程可解析处理的非线性参数化试探函数。作为神经Galerkin方案的前导，我们提出了演化方程具有闭式解的试探函数，重点关注金融领域相关的时变HJB方程。