Error analysis for physics informed neural networks (PINNs) approximating Kolmogorov PDEs
METADATA ONLY
Loading...
Author / Producer
Date
2021-06
Publication Type
Report
ETH Bibliography
yes
Citations
Altmetric
METADATA ONLY
Data
Rights / License
Abstract
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 L2-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.
Permanent link
Publication status
published
Editor
Book title
Journal / series
Volume
2021-17
Pages / Article No.
Publisher
Seminar for Applied Mathematics, ETH Zurich
Event
Edition / version
Methods
Software
Geographic location
Date collected
Date created
Subject
Deep learning; Neural networks; PINNs; Kolmogonov PDE
Organisational unit
03851 - Mishra, Siddhartha / Mishra, Siddhartha
Notes
Funding
Related publications and datasets
Is previous version of: