Core Concepts
The authors present a structure-preserving Eulerian algorithm for solving L2-gradient flows and a structure-preserving Lagrangian algorithm for solving generalized diffusions, both of which employ neural networks as tools for spatial discretization. The proposed schemes are constructed based on the energy-dissipation law directly, guaranteeing the monotonic decay of the system's free energy and ensuring long-term stability of numerical computations.
Abstract
The paper introduces a novel numerical framework, called Energetic Variational Neural Network (EVNN), for solving gradient flows and generalized diffusions. The key ideas are:
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Eulerian EVNN for L2-gradient flows:
- Constructs a finite-dimensional approximation to the continuous energy-dissipation law by introducing a neural network-based spatial discretization.
- Performs temporal discretization before spatial discretization to overcome challenges arising from nonlinear neural network discretization.
- Formulates the update of the neural network parameters as a minimizing movement scheme that guarantees the monotonic decay of the discrete free energy.
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Lagrangian EVNN for generalized diffusions:
- Views the generalized diffusion as an L2-gradient flow of the flow map in the space of diffeomorphisms.
- Seeks an optimal flow map between consecutive time steps, rather than the full flow map, to improve computational efficiency.
- Parameterizes the flow map using neural networks, specifically the convex potential flow architecture, to ensure the map is a diffeomorphism.
The proposed EVNN methods are mesh-free and can solve high-dimensional gradient flows. Numerical experiments demonstrate the accuracy and energy stability of the EVNN schemes.