An overview on deep learning-based approximation methods for partial differential equations
C Beck, M Hutzenthaler, A Jentzen… - arXiv preprint arXiv …, 2020 - arxiv.org
It is one of the most challenging problems in applied mathematics to approximatively solve
high-dimensional partial differential equations (PDEs). Recently, several deep learning …
high-dimensional partial differential equations (PDEs). Recently, several deep learning …
[HTML][HTML] Numerical scheme and stability analysis of stochastic Fitzhugh–Nagumo model
This article deals with the Fitzhugh–Nagumo equation in the presence of stochastic function.
A numerical scheme has been developed for the solution of such equations which preserves …
A numerical scheme has been developed for the solution of such equations which preserves …
Designing universal causal deep learning models: The case of infinite-dimensional dynamical systems from stochastic analysis
Causal operators (CO), such as various solution operators to stochastic differential
equations, play a central role in contemporary stochastic analysis; however, there is still no …
equations, play a central role in contemporary stochastic analysis; however, there is still no …
Multilevel Picard approximations overcome the curse of dimensionality in the numerical approximation of general semilinear PDEs with gradient-dependent …
Neufeld and Wu (arXiv: 2310.12545) developed a multilevel Picard (MLP) algorithm which
can approximately solve general semilinear parabolic PDEs with gradient-dependent …
can approximately solve general semilinear parabolic PDEs with gradient-dependent …
Multilevel Picard algorithm for general semilinear parabolic PDEs with gradient-dependent nonlinearities
In this paper we introduce a multilevel Picard approximation algorithm for general semilinear
parabolic PDEs with gradient-dependent nonlinearities whose coefficient functions do not …
parabolic PDEs with gradient-dependent nonlinearities whose coefficient functions do not …
Differentiable physics: A position piece
B Ramsundar, D Krishnamurthy… - arXiv preprint arXiv …, 2021 - arxiv.org
Differentiable physics provides a new approach for modeling and understanding the
physical systems by pairing the new technology of differentiable programming with classical …
physical systems by pairing the new technology of differentiable programming with classical …
Rectified deep neural networks overcome the curse of dimensionality in the numerical approximation of gradient-dependent semilinear heat equations
Numerical experiments indicate that deep learning algorithms overcome the curse of
dimensionality when approximating solutions of semilinear PDEs. For certain linear PDEs …
dimensionality when approximating solutions of semilinear PDEs. For certain linear PDEs …
Geometry-aware neural solver for fast Bayesian calibration of brain tumor models
Modeling of brain tumor dynamics has the potential to advance therapeutic planning.
Current modeling approaches resort to numerical solvers that simulate the tumor …
Current modeling approaches resort to numerical solvers that simulate the tumor …
Physics-informed neural networks with parameter asymptotic strategy for learning singularly perturbed convection-dominated problem
F Cao, F Gao, X Guo, D Yuan - Computers & Mathematics with Applications, 2023 - Elsevier
Physics-informed neural networks (PINN) have proven their effectiveness in solving partial
differential equations (PDEs). Nevertheless, existing networks cannot model finely detailed …
differential equations (PDEs). Nevertheless, existing networks cannot model finely detailed …
Full error analysis of the random deep splitting method for nonlinear parabolic PDEs and PIDEs
In this paper, we present a randomized extension of the deep splitting algorithm introduced
in Beck et al.(2021) using random neural networks suitable to approximately solve both high …
in Beck et al.(2021) using random neural networks suitable to approximately solve both high …