Error analysis for physics-informed neural networks (PINNs) approximating Kolmogorov PDEs

• Published in: Advances in computational mathematics Vol. 48; no. 6
• Main Authors: De Ryck, Tim, Mishra, Siddhartha
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.12.2022; Springer Nature B.V
• Subjects: Approximation; Black-Scholes equation; Computational mathematics; Computational Mathematics and Numerical Analysis; Computational Science and Engineering; Error analysis; Mathematical and Computational Biology; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Neural networks; Parabolic differential equations; Partial differential equations; Thermodynamics; Training; Visualization; Deep learning; Kolmogorov equations; 65M99; Physics-informed neural networks
• ISSN: 1019-7168; 1572-9044
• DOI: 10.1007/s10444-022-09985-9

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.