M$^3$: Reframing Training Measures for Discretized Physical Simulations
For practitioners of neural operator learning in physical simulations, M³ addresses data distribution bias, enabling data-efficient and physically consistent modeling.
Neural surrogate models for physical simulations suffer from bias due to uneven supervision from discretized training data. The proposed M³ framework balances training measures across scales, achieving up to 4.7× lower error in large-scale volumetric cases and 3–4× reduction in physics-weighted error under aggressive subsampling.
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.