Only Linear Constraints Survive Coarse-Graining: Evaluating Physics-Constrained Neural Operators on Stochastically Forced Turbulence
Abstract
A physics-informed loss adds a discretised PDE residual to the data term, but is only strictly valid if the equations being enforced are closed on the fields being fitted. This fails on coarse-grained, stochastically forced data: nonlinear terms in the PDE generally do not commute with the filter, so the coarse-grained fields do not satisfy the original equations, and driving their residual to zero encodes an implicit closure into the learned operator. Under normalised, translation-invariant filtering on a periodic domain, constant-coefficient linear constraints remain valid on the filtered grid. For the 2D incompressible Navier-Stokes equations, two such constraints are continuity and global momentum balance. We enforce these exactly by applying a closed-form projection on the output of a Fourier neural operator (FNO) at O(N log N) cost. On two-dimensional isotropic turbulence, truncated so that its forcing lies entirely beyond the cutoff wavenumber, the projection achieves a continuity error of 5.0×10−7 against 0.51 for a plain FNO and 0.17 for a physics-informed FNO, and a global momentum balance error of 8.6×10−11 against 7.5×10−4 and 8.3×10−4. The projection only costs an extra 2% of training time, and takes the fraction of rollouts whose energy remains bounded at 448 steps from 0.18 to 0.87. On resolved Kolmogorov flow, test-time optimisation costs approximately 3,300× as much per trajectory as the projection, while leaving maximum divergence three orders of magnitude higher. Nonlinear constraints can also be enforced once their unclosed terms are modelled: closing the global energy balance with a constant subgrid flux, fitted from the coarse training data alone, removes the 20.9% energy deficit that enforcing the unclosed balance produces and keeps every free-running rollout bounded over 2,000 steps.
Related
- On the Non-Negativity of Longitudinal Velocity Autocorrelations in Homogeneous Isotropic Turbulence: Kinematic Constraints, Convexity, and Spectral Broadening
- Direct detection constraints on EFT operators for inelastic dark matter
- Planck Constraints on Turbulence in the Coma Cluster
- Ontological Constraints on Epistemic Syntax: How Informationally-Constrained Access to Reality Shaped the Axiomatic Structure of Quantum, Relativistic, and Gravitational Theories
- Evaluating Proposed Spatiotemporal Matching for Scope 2 Market-based Accounting: emissions, cost, and policy tradeoffs for hourly matching and tighter geographic constraints
BibTeX
@misc{groom2026only,
title = {{Only Linear Constraints Survive Coarse-Graining: Evaluating Physics-Constrained Neural Operators on Stochastically Forced Turbulence}},
author = {Michael Groom and Rafael Oliveira},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.07811},
eprint = {2610.07811},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.07811},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Only Linear Constraints Survive Coarse-Graining: Evaluating Physics-Constrained Neural Operators on Stochastically Forced Turbulence
AU - Michael Groom
AU - Rafael Oliveira
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.07811
UR - https://arxiv.org/abs/2610.07811
AB - A physics-informed loss adds a discretised PDE residual to the data term, but is only strictly valid if the equations being enforced are closed on the fields being fitted. This fails on coarse-grained, stochastically forced data: nonlinear terms in the PDE generally do not commute with the filter, so the coarse-grained fields do not satisfy the original equations, and driving their residual to zero encodes an implicit closure into the learned operator. Under normalised, translation-invariant filtering on a periodic domain, constant-coefficient linear constraints remain valid on the filtered grid. For the 2D incompressible Navier-Stokes equations, two such constraints are continuity and global momentum balance. We enforce these exactly by applying a closed-form projection on the output of a Fourier neural operator (FNO) at O(N log N) cost. On two-dimensional isotropic turbulence, truncated so that its forcing lies entirely beyond the cutoff wavenumber, the projection achieves a continuity error of 5.0×10^−7 against 0.51 for a plain FNO and 0.17 for a physics-informed FNO, and a global momentum balance error of 8.6×10^−11 against 7.5×10^−4 and 8.3×10^−4. The projection only costs an extra 2% of training time, and takes the fraction of rollouts whose energy remains bounded at 448 steps from 0.18 to 0.87. On resolved Kolmogorov flow, test-time optimisation costs approximately 3,300× as much per trajectory as the projection, while leaving maximum divergence three orders of magnitude higher. Nonlinear constraints can also be enforced once their unclosed terms are modelled: closing the global energy balance with a constant subgrid flux, fitted from the coarse training data alone, removes the 20.9% energy deficit that enforcing the unclosed balance produces and keeps every free-running rollout bounded over 2,000 steps.
ER -