M3: Reframing Training Measures for Discretized Physical Simulations

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 M3 (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, M3 consistently improves predictions in the continuous physical domain, achieving up to 4.7× lower error in large-scale volumetric cases. These gains persist under aggressive subsampling (160M rightarrow 16M rightarrow 1.6M points), where M3-trained models outperform those trained on higher-resolution data, reducing physics-weighted relative L2 error by 3--4× and the corresponding MSE by up to 13×. These results highlight data distribution as a key factor in operator learning and position M3 as a scalable, data-efficient approach for physically consistent modeling. Code is available at https://github.com/PhysDataRefine/M3.