Rethinking Metrics for Physics Simulations: Semigroup Error Takes Center Stage

learned physics simulators, traditional evaluation metrics often fail to capture the nuances of long-term prediction accuracy. The focus has been primarily on one-step or short-horizon prediction error. However, these metrics can overlook significant failures in temporal composition and long-horizon rollout. Enter the concept of semigroup error, a promising new diagnostic tool aimed at addressing these shortcomings.

Introducing Semigroup Error

For autonomous systems that are state-complete, exact solution maps should satisfy a semigroup law. This means that evolving a system over two time periods, say s and t, should yield the same result whether done sequentially or in one go over s+t. The proposed normalized semigroup error acts as a post hoc, model-agnostic diagnostic, comparing these direct and composed learned predictions.

Why It Matters

Empirical tests have shown that semigroup error is positively associated with rollout degradation. In experiments with one-dimensional heat and Burgers dynamics using time-conditioned ConvNets and FNO baselines, the trajectory-level Spearman correlation was a notable 0.635, with a 95% confidence interval of [0.621, 0.649]. This indicates that semigroup error is indeed a meaningful measure of long-term prediction quality. But the key finding is that semigroup regularization has mixed effects. It's primarily useful as an evaluation diagnostic rather than a universally beneficial training objective.

Is Semigroup the Silver Bullet?

So, is semigroup error the ultimate solution to the problem of long-term prediction accuracy in learned physics simulators? Not quite. While it provides a valuable lens through which to assess system performance, it doesn't necessarily enhance training outcomes across the board. It's a tool for evaluation, not a magic bullet for training improvement.

This brings us back to the question: What makes a truly effective evaluation metric? Crucially, it should be able to reveal hidden failures that emerge over extended timeframes. Semigroup error does this, but the journey to perfect long-term predictions is far from over. Researchers and practitioners will need to continue to refine both their models and their metrics.

In the quest for more accurate physics simulators, semigroup error represents a significant step forward. However, it should be seen as part of a broader toolkit rather than a standalone solution. After all, in complex systems, one metric rarely tells the whole story.