Estimates on the generalization error of physics-informed neural networks for approximating a class of inverse problems for PDEs
By Siddhartha Mishra, Roberto Molinaro. IMA Journal of Numerical Analysis, 2020
357 citations recorded by the source corpus.
Physics-informed neural networks (PINNs) have recently been very successfully applied for efficiently approximating inverse problems for partial differential equations (PDEs). We focus on a particular class of inverse problems, the so-called data assimilation or unique continuation problems, and prove rigorous estimates on the generalization error of PINNs approximating them. An abstract framework is presented and conditional stability estimates for the underlying inverse problem are employed to derive the estimate on the PINN generalization error, providing rigorous justification for the use of PINNs in this context. The abstract framework is illustrated with examples of four prototypical linear PDEs. Numerical experiments, validating the proposed theory, are also presented.
Subfields: Electronic, Optical and Magnetic Materials, Mathematical Physics, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/86e66da9c6d9928a2e542e1814a5d9f362816d36
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In Electronic, Optical and Magnetic Materials
- Zhiping Mao
- G. Em Karniadakis
- Zhicheng Wang
- Maizar Raissi
- Fabio Giampaolo
- Salvatore Cuomo
- Francesco Piccialli
- Gianluigi Rozza
In Mathematical Physics
In Statistical and Nonlinear Physics
- G. Em Karniadakis
- Zhiping Mao
- Maizar Raissi
- Shengze Cai
- Gianluigi Rozza
- Fabio Giampaolo
- Salvatore Cuomo
- Francesco Piccialli
Citations in the corpus
- Scientific Machine Learning Through Physics–Informed Neural Networks: Where we are and What’s Next (2022, methodology): “The reader can find an operator based mathematical formulation of equation (1) in the work of Mishra and Molinaro (2021a).”
- Physics-informed neural networks (PINNs) for fluid mechanics: a review (2021, background): “[40,41] that the upper bound of the generalization error of PINNs can be quantified, which is dependent on the training error, the number of training samples and some constants from the stability estimate.”