M^3: Reframing Training Measures for Discretized Physical Simulations

arXiv:2605.08843v1 Announce Type: new 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 4.7imes 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--4imes and the corresponding MSE by up to 13imes. 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.

Source: arXiv cs.AI | 2026-05-12