Stable Resolution-Invariant Emulation in Entropically Controlled Kinetic Methods

Resolution invariance is a critical capability for scientific machine learning methods (SciML), however, the ability to sample at any resolution does not guarantee stable emulation at all resolutions. This important distinction underpins one of scientific machine learning's greatest potential: the ability to efficiently emulate complicated and computationally expensive systems on inexpensive coarse grids. Stable numerical solvers are computationally expensive as they attempt to resolve small scales that receive and dissipate spectral energy. Learned coarse-grid forecasters remove the dissipative channel, allowing unresolved frequencies to re-emerge as grid-scale noise (aliasing). We explore stable resolution invariant emulation through physically principled control over entropy dynamics to reinforce the dissipative mechanism within the Neural Discrete Equilibrium (NeurDE) method. Entropically-stabilized NeurDE produces stable long-time rollouts for thousands of forecasting steps on large Reynolds number turbulent flows (Re=50,000) with very coarse grids ($\mathbb{R}^{16\times16}$). Entropically-stabilized NeurDE demonstrates extreme stability at coarse resolutions, outperforming the data-generating stabilized numerical method: a much more challenging baseline than SciML surrogates.