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

Researchers introduce M³ (Multi-scale Morton Measure), a framework that improves neural surrogate models for physical simulations by addressing training bias from discretized data sampling. The method achieves up to 4.7× error reduction in volumetric cases and maintains superior performance even with 90% data reduction, demonstrating that data distribution strategy significantly impacts operator learning efficiency.

M³ addresses a fundamental challenge in training neural surrogate models for physical simulations: the bias introduced when continuous physical domains are represented through discretized sampling. Traditional approaches treat all sampled points equally during training, creating uneven supervision that leads to spatial inconsistencies in physical predictions. This research reframes how training measures are weighted by partitioning space according to physical variation and distributing supervision across multiple scales, achieving measurable improvements in prediction accuracy.

The significance extends beyond academic advancement. Physical simulation surrogate models are increasingly deployed in engineering, climate modeling, materials science, and computational fluid dynamics—domains where model accuracy directly impacts design decisions and resource allocation. Current limitations in data efficiency force practitioners to either collect massive datasets or accept lower-fidelity predictions. M³'s ability to maintain superior results with significantly less data (1.6M points versus 160M) represents a practical breakthrough for resource-constrained applications.

The framework's performance under aggressive subsampling is particularly noteworthy: models trained with M³ on reduced datasets outperform baseline models trained on ten times more data. This suggests that data distribution quality matters more than quantity in operator learning, a principle with broad implications for machine learning in scientific computing. The consistent improvements across diverse industrial-scale datasets with different discretization schemes indicate robustness and generalizability.

Looking forward, this work may accelerate adoption of surrogate models in compute-intensive industries by reducing training data requirements and improving reliability. The methodology could influence how practitioners approach dataset curation and training strategies, potentially unlocking new applications where data collection costs previously prohibited surrogate model deployment.

• →M³ reduces physics prediction error by up to 4.7× in large-scale volumetric simulations through improved training measure balancing
• →The framework maintains superior accuracy with 90% less training data, suggesting data distribution quality outweighs quantity in operator learning
• →Multi-scale partitioning according to physical variation eliminates spatial inconsistencies in neural surrogate model predictions
• →Results persist across diverse industrial datasets with different discretization schemes, indicating strong generalizability
• →The approach positions data efficiency as a key lever for deploying surrogate models in resource-constrained scientific computing applications