Journal: Acta Numerica

Abbreviation

Acta numer.

Publisher

Cambridge University Press

Journal Volumes

ISSN

0962-4929
1474-0508

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Publications 1 - 6 of 6
  • Sparse tensor discretizations of high-dimensional parametric and stochastic PDEs
    Item type: Journal Article
    Schwab, Christoph; Gittelson, Claude Jeffrey (2011)
    Acta Numerica
  • Lanczos-type solvers for nonsymmetric linear systems of equations
    Item type: Journal Article
    Gutknecht, Martin H. (1997)
    Acta Numerica
  • Numerical analysis of physics-informed neural networks and related models in physics-informed machine learning
    Item type: Journal Article
    De Ryck, Tim; Mishra, Siddhartha (2024)
    Acta Numerica
    Physics-informed neural networks (PINNs) and their variants have been very popular in recent years as algorithms for the numerical simulation of both forward and inverse problems for partial differential equations. This article aims to provide a comprehensive review of currently available results on the numerical analysis of PINNs and related models that constitute the backbone of physics-informed machine learning. We provide a unified framework in which analysis of the various components of the error incurred by PINNs in approximating PDEs can be effectively carried out. We present a detailed review of available results on approximation, generalization and training errors and their behaviour with respect to the type of the PDE and the dimension of the underlying domain. In particular, we elucidate the role of the regularity of the solutions and their stability to perturbations in the error analysis. Numerical results are also presented to illustrate the theory. We identify training errors as a key bottleneck which can adversely affect the overall performance of various models in physics-informed machine learning.
  • Finite elements in computational electromagnetism
    Item type: Journal Article
    Hiptmair, R. (2002)
    Acta Numerica
  • Numerical methods with controlled dissipation for small-scale dependent shocks
    Item type: Journal Article
    LeFloch, Philippe G.; Mishra, Siddhartha (2014)
    Acta Numerica
  • On the computation of measure-valued solutions
    Item type: Journal Article
    Fjordholm, Ulrik S.; Mishra, Siddhartha; Tadmor, Eitan (2016)
    Acta Numerica
    A standard paradigm for the existence of solutions in fluid dynamics is based on the construction of sequences of approximate solutions or approximate minimizers. This approach faces serious obstacles, most notably in multi-dimensional problems, where the persistence of oscillations at ever finer scales prevents compactness. Indeed, these oscillations are an indication, consistent with recent theoretical results, of the possible lack of existence/uniqueness of solutions within the standard framework of integrable functions. It is in this context that Young measures – parametrized probability measures which can describe the limits of such oscillatory sequences – offer the more general paradigm of measure-valued solutions for these problems. We present viable numerical algorithms to compute approximate measure-valued solutions, based on the realization of approximate measures as laws of Monte Carlo sampled random fields. We prove convergence of these algorithms to measure-valued solutions for the equations of compressible and incompressible inviscid fluid dynamics, and present a large number of numerical experiments which provide convincing evidence for the viability of the new paradigm. We also discuss the use of these algorithms, and their extensions, in uncertainty quantification and contexts other than fluid dynamics, such as non-convex variational problems in materials science.
Publications 1 - 6 of 6