Symplectic Neural Flows for Modeling and Discovery
This work addresses the need for reliable long-term simulations in physics and engineering by providing a neural network-based method that incorporates symplectic principles, though it appears incremental in applying neural networks to an underexplored area of geometric integration.
The paper tackled the problem of modeling complex physical systems by introducing SympFlow, a time-dependent symplectic neural network that preserves key properties like energy and momentum, demonstrating improved energy conservation and accuracy in chaotic and dissipative systems compared to general-purpose methods.
Hamilton's equations are fundamental for modeling complex physical systems, where preserving key properties such as energy and momentum is crucial for reliable long-term simulations. Geometric integrators are widely used for this purpose, but neural network-based methods that incorporate these principles remain underexplored. This work introduces SympFlow, a time-dependent symplectic neural network designed using parameterized Hamiltonian flow maps. This design allows for backward error analysis and ensures the preservation of the symplectic structure. SympFlow allows for two key applications: (i) providing a time-continuous symplectic approximation of the exact flow of a Hamiltonian system--purely based on the differential equations it satisfies, and (ii) approximating the flow map of an unknown Hamiltonian system relying on trajectory data. We demonstrate the effectiveness of SympFlow on diverse problems, including chaotic and dissipative systems, showing improved energy conservation compared to general-purpose numerical methods and accurate