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A Deep Double Ritz Method (D2RM) for solving Partial Differential Equations using Neural Networks
- Publication Year :
- 2023
- Publisher :
- Elsevier, 2023.
-
Abstract
- Residual minimization is a widely used technique for solving Partial Differential Equations in variational form. It minimizes the dual norm of the residual, which naturally yields a saddle-point (min-max) problem over the so-called trial and test spaces. In the context of neural networks, we can address this min-max approach by employing one network to seek the trial minimum, while another network seeks the test maximizers. However, the resulting method is numerically unstable as we approach the trial solution. To overcome this, we reformulate the residual minimization as an equivalent minimization of a Ritz functional fed by optimal test functions computed from another Ritz functional minimization. We call the resulting scheme the Deep Double Ritz Method (D$^2$RM), which combines two neural networks for approximating trial functions and optimal test functions along a nested double Ritz minimization strategy. Numerical results on different diffusion and convection problems support the robustness of our method, up to the approximation properties of the networks and the training capacity of the optimizers.<br />28 pages
- Subjects :
- FOS: Computer and information sciences
variational formulation
Computer Science - Machine Learning
residual minimization
optimal test functions
FOS: Mathematics
partial differential equations
Numerical Analysis (math.NA)
Mathematics - Numerical Analysis
neural networks
Ritz method
Machine Learning (cs.LG)
Subjects
Details
- Language :
- English
- Database :
- OpenAIRE
- Accession number :
- edsair.doi.dedup.....6949e597eb7f4804407a5caf0ceb83b5