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Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEs II: A class of inverse problems
(2020)SAM Research ReportPhysics informed neural networks (PINNs) have recently been very successfully applied for efficiently approximating inverse problems for 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 ...Report -
Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEs
(2020)SAM Research ReportPhysics informed neural networks (PINNs) have recently been widely used for robust and accurate approximation of PDEs. We provide rigorous 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 ...Report -
Physics Informed Neural Networks for Simulating Radiative Transfer
(2020)SAM Research ReportWe propose a novel machine learning algorithm for simulating radiative transfer. Our algorithmis based on physics informed neural networks (PINNs), which are trained by minimizing the residualof the underlying radiative tranfer equations. We present extensive experiments and theoretical errorestimates to demonstrate that PINNs provide a very easy to implement, fast, robust and accuratemethod for simulating radiative transfer. We also ...Report