M$^3$: Reframing Training Measures for Discretized Physical Simulations

TL;DR

M$^3$ is a scalable framework that improves neural surrogate models for physical simulations by balancing training data across multiple scales, leading to more accurate and consistent predictions in discretized domains.

Contribution

The paper introduces M$^3$, a novel multi-scale measure balancing method that enhances physical fidelity in neural surrogate models trained on discretized data.

Findings

M$^3$ reduces prediction error by up to 4.7× in large-scale volumetric simulations.

Models trained with M$^3$ outperform higher-resolution data models under subsampling.

Data distribution significantly impacts operator learning accuracy.

Abstract

Neural surrogate models for physical simulations are trained on discretized samples of continuous domains, where the induced empirical measure leads to uneven supervision, biasing optimization and causing spatial inconsistencies in physical fidelity. To mitigate this measure-induced bias, we propose M$^3$ (Multi-scale Morton Measure), a scalable framework that balances training measures by partitioning space according to physical variation and allocating supervision across multiple scales. Applied to three industrial-scale datasets with diverse discretizations, M$^3$ consistently improves predictions in the continuous physical domain, achieving up to 4.7$\times$ lower error in large-scale volumetric cases. These gains persist under aggressive subsampling (160M $\rightarrow$ 16M $\rightarrow$ 1.6M points), where M$^3$-trained models outperform those trained on higher-resolution data, reducing physics-weighted relative $L_2$ error by 3--4$\times$ and the corresponding MSE by up to 13$\times$. These results highlight data distribution as a key factor in operator learning and position M$^3$ as a scalable, data-efficient approach for physically consistent modeling.