Estimates on the generalization error of physics-informed neural networks for approximating PDEs
Siddhartha Mishra, Roberto Molinaro
Abstract
Abstract Physics-informed neural networks (PINNs) have recently been widely used for robust and accurate approximation of partial differential equations (PDEs). We provide upper bounds on the generalization error of PINNs approximating solutions of the forward problem for PDEs. An abstract formalism is introduced and stability properties of the underlying PDE are leveraged to derive an estimate for the generalization error in terms of the training error and number of training samples. This abstract framework is illustrated with several examples of nonlinear PDEs. Numerical experiments, validating the proposed theory, are also presented.
Topics & Concepts
GeneralizationPartial differential equationMathematicsNonlinear systemApplied mathematicsFormalism (music)Artificial neural networkError analysisApproximation errorGeneralization errorStability (learning theory)Computer scienceMathematical analysisArtificial intelligenceMachine learningArtPhysicsVisual artsQuantum mechanicsMusicalModel Reduction and Neural NetworksMagnetic Properties and ApplicationsNeural Networks and Applications