Structure-Preserving Operator Learning

TL;DR

This work introduces a family of operator learning architectures, structure-preserving operator networks (SPONs), that allows to preserve key mathematical and physical properties of the continuous system by leveraging finite element discretizations of the input-output spaces.

Abstract

Learning complex dynamics driven by partial differential equations directly from data holds great promise for fast and accurate simulations of complex physical systems. In most cases, this problem can be formulated as an operator learning task, where one aims to learn the operator representing the physics of interest, which entails discretization of the continuous system. However, preserving key continuous properties at the discrete level, such as boundary conditions, and addressing physical systems with complex geometries is challenging for most existing approaches. We introduce a family of operator learning architectures, structure-preserving operator networks (SPONs), that allows to preserve key mathematical and physical properties of the continuous system by leveraging finite element (FE) discretizations of the input-output spaces. SPONs are encode-process-decode architectures that are end-to-end differentiable, where the encoder and decoder follows from the discretizations of the input-output spaces. SPONs can operate on complex geometries, enforce certain boundary conditions exactly, and offer theoretical guarantees. Our framework provides a flexible way of devising structure-preserving architectures tailored to specific applications, and offers an explicit trade-off between performance and efficiency, all thanks to the FE discretization of the input-output spaces. Additionally, we introduce a multigrid-inspired SPON architecture that yields improved performance at higher efficiency. Finally, we release a software to automate the design and training of SPON architectures.

Figures (11)

• Figure 1: Schematic diagram of a structure-preserving operator network architecture.
• Figure 2: Diagram of the multigrid processor $\mathcal{P}_{\theta}^{MG}$ for 3 levels (right) with the corresponding mesh hierarchy (left). $\mathcal{P}_{\theta}^{MG}$ takes in $\bar{f}$, the input DoFs, and predicts $\bar{u}$, the DoFs of the output.
• Figure 3: Left: Relative errors across the epochs for different benchmarks (trained and tested at $n_{x} = 64$). Middle: Relative errors of SPON and SPON-MG trained at $n_{x} = 64$ and evaluated at different resolutions Right: Relative errors when trained and evaluated at different resolutions.
• Figure 4: Left: Exact fluid velocity flow and magnitude from a random source in the test set (left boundary condition) at different time steps in the simulation. Right: Predicted velocity solution from the same initial condition at $t=2.2$. The bottom row (highlighted in blue) is an extrapolation test as the time step has not been observed in training.
• Figure 5: Diagram illustrating the graph induced by a continuous Lagrange finite element of order 1 ($\mathrm{CG}_1$) (left) and order 2 ($\mathrm{CG}_2$) (right) on a square uniform and structured mesh with triangles. The neighbors (blue) of a given node (red) are illustrated. The latent graph's edges can be derived by finding the neighbors of each DoF using \ref{['eq:latent_graph_definition_appendix']}. Note that the graph's edges do not necessarily coincide with the mesh edges. In this example, for the $\mathrm{CG}_1$ (resp. $\mathrm{CG}_2$) discretization, each node has at most 7 (resp. 19) edges in the latent graph.

Theorems & Definitions (2)

• Theorem 1: Approximation bound
• proof : Proof of \ref{['thm_approximation_error']}