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We present a deep learning algorithm for the numerical solution of parametricfamilies of high-dimensional linear Kolmogorov partial differential equations(PDEs). Our method is based on reformulating the numerical approximation of awhole family of Kolmogorov PDEs as a single statistical learning problem usingthe Feynman-Kac formula. Successful numerical experiments are presented, whichempirically confirm the functionality and efficiency of our proposed algorithmin the case of heat equations and Black-Scholes option pricing modelsparametrized by affine-linear coefficient functions. We show that a single deepneural network trained on simulated data is capable of learning the solutionfunctions of an entire family of PDEs on a full space-time region. Mostnotably, our numerical observations and theoretical results also demonstratethat the proposed method does not suffer from the curse of dimensionality,distinguishing it from almost all standard numerical methods for PDEs.

Numerically Solving Parametric Families of High-Dimensional Kolmogorov Partial Differential Equations via Deep Learning