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

2026-05-09

AI Summary

This work addresses the challenge in discretized physics simulation where neural surrogate models often suffer from optimization bias and spatial inconsistency in physical fidelity due to non-uniform empirical measures. To resolve this, the authors propose the M³ framework, which uniquely integrates multiscale Morton measure–driven spatial partitioning with measure reweighting. By combining spatially adaptive sampling, a physics-weighted loss function, and an operator learning architecture, M³ allocates supervisory signals across multiple scales to balance the training distribution. The approach significantly enhances prediction consistency over continuous physical domains and improves data efficiency: it reduces error by up to 4.7× in large-scale voxelized cases and achieves 3–4× lower physics-weighted L₂ error under extreme downsampling (from 160M to 1.6M points), with mean squared error decreasing by as much as 13×.

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.

Problem

• discretized physical simulations
• empirical measure bias
• spatial inconsistency
• physical fidelity
• operator learning

Innovation

• Multi-scale Morton Measure
• measure-induced bias
• neural surrogate models
• operator learning
• physically consistent modeling