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 $L^p$ 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.