Error analysis for physics informed neural networks (PINNs) approximating Kolmogorov PDEs
This provides a theoretical foundation for PINNs in solving high-dimensional linear parabolic PDEs, which is incremental as it extends error analysis to a specific class of equations.
The paper tackles the error analysis of Physics Informed Neural Networks (PINNs) for approximating solutions to Kolmogorov PDEs, such as the heat and Black-Scholes equations, by deriving rigorous bounds on the generalization and total L^2-error, and proving that network size and training samples grow polynomially with dimension to avoid the curse of dimensionality.
Physics informed neural networks approximate solutions of PDEs by minimizing pointwise residuals. We derive rigorous bounds on the error, incurred by PINNs in approximating the solutions of a large class of linear parabolic PDEs, namely Kolmogorov equations that include the heat equation and Black-Scholes equation of option pricing, as examples. We construct neural networks, whose PINN residual (generalization error) can be made as small as desired. We also prove that the total $L^2$-error can be bounded by the generalization error, which in turn is bounded in terms of the training error, provided that a sufficient number of randomly chosen training (collocation) points is used. Moreover, we prove that the size of the PINNs and the number of training samples only grow polynomially with the underlying dimension, enabling PINNs to overcome the curse of dimensionality in this context. These results enable us to provide a comprehensive error analysis for PINNs in approximating Kolmogorov PDEs.