发表机构
University of Maryland, College Park（马里兰大学帕克分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明有限表达式符号表示类可克服高维PDE的维数灾难，通过多级皮卡德框架传播及构造确定性逼近，实现多项式复杂度保证，为有限表达式方法奠定理论基础。
AI 中文摘要
我们证明了有限表达式构成一种符号表示类，能够克服若干类高维偏微分方程的维数灾难。对于半线性热方程，我们展示了如何将终端条件和非线性项的有限表达式逼近通过多级皮卡德框架传播，从而产生具有指定均方根精度的随机逐点逼近。对于具有拉普拉斯扩散和零漂移的半线性柯尔莫哥洛夫方程、对角布莱克-舒尔斯方程以及半空间上的拉普拉斯狄利克雷问题，我们构造了具有任意小空间$L^p$误差的确定性有限表达式逼近。在适当的增长和正则性假设下，所得逼近的评估成本在维数和倒数精度上以多项式为界。非线性情形的一个关键要素是构造逼近PDE解的有限表达式，同时保留随机解理论所需的增长和Lipschitz结构。更广泛地，我们的结果表明，高维PDE的维度鲁棒逼近并不局限于传统的神经网络架构：从固定字典生成的结构化符号表达式可以实现可比较的多项式复杂度保证。这为有限表达式方法作为高维科学计算的一种表示范式提供了严格的基础。
英文摘要
We establish that finite expressions form a symbolic representation class capable of overcoming the curse of dimensionality for several classes of high-dimensional partial differential equations. For semilinear heat equations, we show how finite expression approximations of the terminal condition and nonlinearity can be propagated through the multilevel Picard framework to produce randomized pointwise approximations with prescribed root-mean-square accuracy. For semilinear Kolmogorov equations with Laplacian diffusion and zero drift, diagonal Black--Scholes equations, and the Laplace Dirichlet problem on a half-space, we construct deterministic finite expression approximations with arbitrarily small spatial $L^p$ error. Under suitable growth and regularity assumptions, the evaluation cost of the resulting approximants is bounded polynomially in the dimension and the reciprocal accuracy. A key ingredient in the nonlinear setting is the construction of finite expressions that approximate the PDE solution while preserving the growth and Lipschitz structures required by the stochastic solution theory. More broadly, our results show that dimension-robust approximation of high-dimensional PDEs is not restricted to conventional neural-network architectures: structured symbolic expressions generated from a fixed dictionary can achieve comparable polynomial-complexity guarantees. This provides a rigorous foundation for finite expression methods as a representation paradigm for high-dimensional scientific computing.
Comments26 pages
