Journal
SIAM JOURNAL ON SCIENTIFIC COMPUTING
Volume 43, Issue 5, Pages A3135-A3154Publisher
SIAM PUBLICATIONS
DOI: 10.1137/19M1297919
Keywords
splitting-up method; neural networks; deep learning; nonlinear partial differential equations
Categories
Funding
- Swiss National Science Foundation [200020 175699]
- Deutsche Forschungsgemeinschaft [EXC 2044-390685587]
- Nanyang Assistant Professorship grant Machine Learning based Algorithms in Finance and Insurance
- Swiss National Science Foundation (SNF) [200020_175699] Funding Source: Swiss National Science Foundation (SNF)
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The method combines operator splitting with deep learning for numerical solutions of nonlinear parabolic PDEs. It divides the problem into separate learning tasks which can handle high dimensional PDEs efficiently. Testing on various examples shows excellent results in up to 10,000 dimensions with short run times.
In this paper, we introduce a numerical method for nonlinear parabolic partial differential equations (PDEs) that combines operator splitting with deep learning. It divides the PDE approximation problem into a sequence of separate learning problems. Since the computational graph for each of the subproblems is comparatively small, the approach can handle extremely high dimensional PDEs. We test the method on different examples from physics, stochastic control, and mathematical finance. In all cases, it yields very good results in up to 10,000 dimensions with short run times.
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