Learning Implicit PDE Integration with Linear Implicit Layers

Neural networks can learn local interactions to faithfully reproduce large-scale dynamics in important physical systems. Trained on PDE integrations or noisy observations, these emulators can assimilate data, tune parameters and learn sub-grid process representations. However, implicit integration schemes cannot be expressed as local feedforward computations. We therefore introduce linear implicit layers (LILs), which learn and solve linear systems with locally computed coefficients. LILs use diagonal dominance to ensure parallel solver convergence and support efficient backward mode differentiation. As a challenging test case, we train emulators on semi-implicit integration of 2D shallow-water equations with closed boundaries. LIL networks learned compact representations of the local interactions controlling the 30.000 degrees of freedom of this discretized system of PDEs. This enabled accurate and stable LIL-based emulation over many time steps where feedforward networks failed.