Hugo Melchers

Hugo Melchers, Daan Crommelin, Barry Koren et al.

Neural closure models have recently been proposed as a method for efficiently approximating small scales in multiscale systems with neural networks. The choice of loss function and associated training procedure has a large effect on the accuracy and stability of the resulting neural closure model. In this work, we systematically compare three distinct procedures: "derivative fitting", "trajectory fitting" with discretise-then-optimise, and "trajectory fitting" with optimise-then-discretise. Derivative fitting is conceptually the simplest and computationally the most efficient approach and is found to perform reasonably well on one of the test problems (Kuramoto-Sivashinsky) but poorly on the other (Burgers). Trajectory fitting is computationally more expensive but is more robust and is therefore the preferred approach. Of the two trajectory fitting procedures, the discretise-then-optimise approach produces more accurate models than the optimise-then-discretise approach. While the optimise-then-discretise approach can still produce accurate models, care must be taken in choosing the length of the trajectories used for training, in order to train the models on long-term behaviour while still producing reasonably accurate gradients during training. Two existing theorems are interpreted in a novel way that gives insight into the long-term accuracy of a neural closure model based on how accurate it is in the short term.

Hugo Melchers, Joost Prins, Michael Abdelmalik

This work introduces a paradigm for constructing parametric neural operators that are derived from finite-dimensional representations of Green's operators for linear partial differential equations (PDEs). We refer to such neural operators as Neural Green's Operators (NGOs). Our construction of NGOs preserves the linear action of Green's operators on the inhomogeneity fields, while approximating the nonlinear dependence of the Green's function on the coefficients of the PDE using neural networks. This construction reduces the complexity of the problem from learning the entire solution operator and its dependence on all parameters to only learning the Green's function and its dependence on the PDE coefficients. Furthermore, we show that our explicit representation of Green's functions enables the embedding of desirable mathematical attributes in our NGO architectures, such as symmetry, spectral, and conservation properties. Through numerical benchmarks on canonical PDEs, we demonstrate that NGOs achieve comparable or superior accuracy to Deep Operator Networks, Variationally Mimetic Operator Networks, and Fourier Neural Operators with similar parameter counts, while generalizing significantly better when tested on out-of-distribution data. For parametric time-dependent PDEs, we show that NGOs that are trained on a single time step can produce pointwise-accurate dynamics in an auto-regressive manner over arbitrarily large numbers of time steps. For parametric nonlinear PDEs, we demonstrate that NGOs trained exclusively on solutions of corresponding linear problems can be embedded within iterative solvers to yield accurate solutions, provided a suitable initial guess is available. Finally, we show that we can leverage the explicit representation of Green's functions returned by NGOs to construct effective matrix preconditioners that accelerate iterative solvers for PDEs.