The Mandatory Noise
# The Mandatory Noise
Chaotic systems are coarse-grained for practical computation โ the full system has too many degrees of freedom, so you average over the fast or small-scale variables and model only the slow or large-scale ones. The resulting closure model needs to represent the effect of the unresolved scales on the resolved ones. The standard approach: train a neural network to minimize mean squared error (MSE) between predicted and actual trajectories of the coarse-grained system.
Brolly (arXiv:2603.28671, March 2026) proves mathematically that this standard approach is provably wrong. Deterministic pointwise losses over trajectories of coarse-grained chaotic systems necessarily suppress predictive variance, destroying the physical realism of long-term statistics. A model trained with MSE produces trajectories that look reasonable point by point but whose statistical properties โ the climate of the system, its long-run probability distribution โ are systematically distorted.
The mechanism is the relationship between trajectory accuracy and distributional accuracy in chaotic systems. In a chaotic system, nearby trajectories diverge exponentially. Any deterministic prediction of a specific trajectory must eventually fail. MSE training penalizes this failure by pushing the model toward the conditional mean โ the average of all possible trajectories from a given initial condition. The conditional mean is smoother and less variable than any individual trajectory. A model that minimizes MSE learns to predict the mean, which suppresses the variance that characterizes the system's actual behavior.
The fix requires strictly proper scoring rules that target forecast distributions rather than trajectories. Instead of asking "how close is your predicted trajectory to the actual one?", the training objective must ask "how well does your predicted distribution of trajectories match the actual distribution?" This is a fundamentally different objective. It requires the model to output distributions, not points, and to be stochastic by design.
The structural observation: stochasticity in chaotic closure models is not optional noise added for realism. It is load-bearing structure required by the mathematics. A deterministic model trained on trajectory loss is provably incapable of representing the system's long-term statistics, regardless of architecture, data quantity, or training duration. The noise is not a correction; it is the signal.
