We propose Neural Walk-on-Spheres (NWoS), a novel neural PDE solver for the
efficient solution of high-dimensional Poisson equations. Leveraging stochastic
representations and Walk-on-Spheres methods, we develop novel losses for neural
networks based on the recursive solution of Poisson equations on spheres inside
the domain. The resulting method is highly parallelizable and does not require
spatial gradients for the loss. We provide a comprehensive comparison against
competing methods based on PINNs, the Deep Ritz method, and (backward)
stochastic differential equations. In several challenging, high-dimensional
numerical examples, we demonstrate the superiority of NWoS in accuracy, speed,
and computational costs. Compared to commonly used PINNs, our approach can
reduce memory usage and errors by orders of magnitude. Furthermore, we apply
NWoS to problems in PDE-constrained optimization and molecular dynamics to show
its efficiency in practical applications.

提出了一种名为 Neural Walk-on-Spheres（NWoS）的新型神经 PDE 求解器，用于高维 Poisson 方程的高效求解。通过利用随机表示和 Walk-on-Spheres 方法，基于球内泊松方程的递归求解，我们开发了用于神经网络的新型损失函数。该方法具有高度的可并行性，且不需要损失函数的空间梯度。通过与基于 PINNs，Deep Ritz 方法和（反向）随机微分方程的竞争方法进行全面比较，在多个具有挑战性的高维数值示例中，我们证明了 NWoS 在准确性、速度和计算成本方面的优越性。与常用的 PINNs 相比，我们的方法可以将内存使用和误差降低数个数量级。此外，我们还将 NWoS 应用于 PDE 约束优化和分子动力学问题中，展示了其在实际应用中的高效性。

使用神经行星行走方法解决泊松方程

Solving Poisson Equations using Neural Walk-on-Spheres

Recently, neural networks have been extensively employed to solve partial
differential equations (PDEs) in physical system modeling. While major studies
focus on learning system evolution on predefined static mesh discretizations,
some methods utilize reinforcement learning or supervised learning techniques
to create adaptive and dynamic meshes, due to the dynamic nature of these
systems. However, these approaches face two primary challenges: (1) the need
for expensive optimal mesh data, and (2) the change of the solution space's
degree of freedom and topology during mesh refinement. To address these
challenges, this paper proposes a neural PDE solver with a neural mesh adapter.
To begin with, we introduce a novel data-free neural mesh adaptor, called
Data-free Mesh Mover (DMM), with two main innovations. Firstly, it is an
operator that maps the solution to adaptive meshes and is trained using the
Monge-Ampere equation without optimal mesh data. Secondly, it dynamically
changes the mesh by moving existing nodes rather than adding or deleting nodes
and edges. Theoretical analysis shows that meshes generated by DMM have the
lowest interpolation error bound. Based on DMM, to efficiently and accurately
model dynamic systems, we develop a moving mesh based neural PDE solver
(MM-PDE) that embeds the moving mesh with a two-branch architecture and a
learnable interpolation framework to preserve information within the data.
Empirical experiments demonstrate that our method generates suitable meshes and
considerably enhances accuracy when modeling widely considered PDE systems.

本文提出了一种利用神经网格适配器的神经 PDE 求解器和基于移动网格的神经 PDE 求解器，以解决传统方法中昂贵优化网格数据和自由度拓扑变化的挑战，并通过实验证明了方法的有效性。

通过无数据网格移动器改进神经 PDE 求解器

Better Neural PDE Solvers Through Data-Free Mesh Movers

Hamilton-Jacobi (HJ) reachability analysis is an important formal
verification method for guaranteeing performance and safety properties of
dynamical control systems. Its advantages include compatibility with general
nonlinear system dynamics, formal treatment of bounded disturbances, and the
ability to deal with state and input constraints. However, it involves solving
a PDE, whose computational and memory complexity scales exponentially with
respect to the number of state variables, limiting its direct use to
small-scale systems. We propose DeepReach, a method that leverages new
developments in sinusoidal networks to develop a neural PDE solver for
high-dimensional reachability problems. The computational requirements of
DeepReach do not scale directly with the state dimension, but rather with the
complexity of the underlying reachable tube. DeepReach achieves comparable
results to the state-of-the-art reachability methods, does not require any
explicit supervision for the PDE solution, can easily handle external
disturbances, adversarial inputs, and system constraints, and also provides a
safety controller for the system. We demonstrate DeepReach on a 9D
multi-vehicle collision problem, and a 10D narrow passage problem, motivated by
autonomous driving applications.

DeepReach 是一种新的方法，利用了正弦网络的新发展，针对高维可达性问题开发了神经偏微分方程求解器，不需要任何显式监督，可以轻松处理外部干扰，对抗性输入和系统约束，并为系统提供安全控制器。在多车碰撞问题和狭窄通道问题上，DeepReach 可以实现与最先进的可达性方法相当的结果，这些问题是受自动驾驶应用启发的。