Solving PDEs by variational physics-informed neural networks: an a posteriori error analysis

Abstract

We consider the discretization of elliptic boundary-value problems by variational physics-informed neural networks (VPINNs), in which test functions are continuous, piecewise linear functions on a triangulation of the domain. We define an a posteriori error estimator, made of a residual-type term, a loss-function term, and data oscillation terms. We prove that the estimator is both reliable and efficient in controlling the energy norm of the error between the exact and VPINN solutions. Numerical results are in excellent agreement with the theoretical predictions.

Enforcing Dirichlet boundary conditions in physics-informed neural networks and variational physics-informed neural networks

“…This implies that there exist other sources of error that dominate and that a very small loss function does not ensure a very accurate solution; this phenomenon is also observed in Fig. 3 of [45] and is discussed in greater detail therein.…”

Robust Variational Physics-Informed Neural Networks

“…A typical definition for the loss functional in (12) found in the literature is the following (see, e.g., [31,33,5,6]):…”

“…where (• , •) 0 denotes the L 2 -inner product 5 , and ⟨• , •⟩ denotes the duality map between V ′ and V , coinciding with ( f , v) 0 when f belongs to L 2 (Ω).…”

“…That is, given a discrete test space, the method's loss, stability, and robustness heavily depends upon the choice of the basis functions of the given test space. In [5], authors present an a posteriori error analysis for discretizing elliptic boundary-value problems with VPINNs employing piecewise polynomials for the test space. As the loss function in VPINNs is, in general, not robust with respect to the true error (for example, the loss function can tend to zero even if the true error does not), they provide an error estimator employing classical techniques in finite element analysis to obtain practical information on the quality of the approximation.…”

Neural control of discrete weak formulations: Galerkin, least squares & minimal-residual methods with quasi-optimal weights

“…In particular, when α > L 1 , where L 1 is the Lipschitz constant of j ′ 1 (•), then γ = α − L 1 20. There are some works on a posteriori error analysis for neural networks approximations; see, e.g.,[63].…”