Finite Expression Approximation of High-Dimensional PDEs Without the Curse of Dimensionality
Abstract
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 Lp 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.
Related
- Asymptotic approximation of sensitivities in finite dimensional continuous data assimilation with application to parameter estimation
- Finite-dimensional normed spaces without small-norm hyperplane projections
- The Approximation Property without the Bounded Approximation Property in a Separable Banach Space and Its Dual
- Visuomotor coordination on the road: Low-dimensional representations reveal adaptive, context-dependent reductions in the dimensionality of natural driving behavior
- Complete non-orientable minimal surfaces of finite total curvature: Approximation and interpolation
BibTeX
@misc{liu2026finite,
title = {{Finite Expression Approximation of High-Dimensional PDEs Without the Curse of Dimensionality}},
author = {Zhi Heng Liu and Haizhao Yang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.32229},
eprint = {2609.32229},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.32229},
}
RIS
TY - UNPB
TI - Finite Expression Approximation of High-Dimensional PDEs Without the Curse of Dimensionality
AU - Zhi Heng Liu
AU - Haizhao Yang
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2609.32229
UR - https://arxiv.org/abs/2609.32229
AB - 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.
ER -