  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.
