Enforcing Energy Conservation in ML-Based Approximations of Nonlinear Four-Wave Interactions

Abstract Accurate and efficient approximation of nonlinear four-wave interactions remains one of the longstanding challenge in spectral wave modeling, as exact calculations are extremely computationally too demanding for practical models. Consequently, most operational wave models employ simplified parameterizations such as the Discrete Interaction Approximation (DIA), for fast computational speed despite known deficiencies. Recent advances in machine learning offer a promising alternative, but standard neural networks do not inherently conserve fundamental physical quantities such as energy, wave action, and momentum, potentially leading to unphysical energy shifts and numerical instability during predictions with these physically inconsistent parameterizations. This study develops two energy-conserving machine learning approaches: a soft constraint that penalizes energy imbalance in the loss function and a hard constraint implemented as a custom network layer that enforces energy conservation within the network architecture. Both approaches substantially reduce energy imbalance compared with unconstrained model, with the hard-constrained approach achieving exact conservation, while also improving numerical stability and generalization to unseen sea states, providing a physically consistent framework for computing nonlinear four-wave interactions.