We present a deep learning algorithm for the numerical solution of parametric families of high-dimensional linear Kolmogorov partial differential equations (PDEs). Our method is based on reformulating the numerical approximation of a whole family of Kolmogorov PDEs as a single statistical learning problem using the Feynman-Kac formula. Successful numerical experiments are presented, which empirically confirm the functionality and efficiency of our proposed algorithm in the case of heat equations and Black-Scholes option pricing models parametrized by affine-linear coefficient functions. We show that a single deep neural network trained on simulated data is capable of learning the solution functions of an entire family of PDEs on a full space-time region. Most notably, our numerical observations and theoretical results also demonstrate that the proposed method does not suffer from the curse of dimensionality, distinguishing it from almost all standard numerical methods for PDEs.

Weaknesses: The authors restrict to affine-linear parameters, but this restriction is not a requirement of the methodology and it is unclear from the text why the restriction was put in place. Is the purpose solely to narrow the scope of the analysis, or is there some performance loss for more general coefficients? It would be useful for the authors to comment in the paper on why this restriction is imposed. Similarly the theoretical results proven apply only to the example from Section 3.3 with a specific initial condition. A more general result would be preferable, particularly since the highest dimension application the authors have provided is actually for a different initial condition (though, of course, the input dimension for the parabaloid initial condition is still rather high). However a more general result may not be straightforward to derive.

The three reviewers, who hail from different sub-communities that all overlap with the paper's content, agree that this is a very well presented work that combines rarely used techniques (such as Feynman-Kac) to interesting ML use cases. It should thus be accepted. The reviewers also raised some concerns about the presentation of the experiments. Please make sure to address these for the camera-ready version.

We present a deep learning algorithm for the numerical solution of parametric families of high-dimensional linear Kolmogorov partial differential equations (PDEs). Our method is based on reformulating the numerical approximation of a whole family of Kolmogorov PDEs as a single statistical learning problem using the Feynman-Kac formula. Successful numerical experiments are presented, which empirically confirm the functionality and efficiency of our proposed algorithm in the case of heat equations and Black-Scholes option pricing models parametrized by affine-linear coefficient functions. We show that a single deep neural network trained on simulated data is capable of learning the solution functions of an entire family of PDEs on a full space-time region. Most notably, our numerical observations and theoretical results also demonstrate that the proposed method does not suffer from the curse of dimensionality, distinguishing it from almost all standard numerical methods for PDEs.