MoNo: Multiscale Optimal Transport Neural Operator for Solving PDEs on General Geometries
2026-08-10 • Machine Learning
Machine LearningArtificial Intelligence
AI summaryⓘ
The authors study how transformer-based neural operators solve PDEs by turning spatial data into compact tokens but find that current methods unevenly assign data to these tokens, causing some to be overloaded and others unused. This imbalance worsens in deeper, hierarchical network layers, leading to a collapse of token usefulness. To fix this, they propose MoNo, which uses a new method called CoTAP that balances assignments between layers using optimal transport theory, ensuring stable and efficient information flow. Their approach improves performance and efficiency in solving PDEs across different geometries.
Partial Differential EquationsNeural OperatorsTransformersLatent SpaceToken AssignmentHierarchical ArchitectureOptimal TransportMultiscale ModelingEntropy RegularizationGeometric Deep Learning
Authors
Zijiang Yang, Xiaomeng Wu, Dongmei Fu
Abstract
Transformer-based neural operators have achieved substantial progress in solving Partial Differential Equations (PDEs) by projecting spatial observations into compact latent tokens and learning physical interactions in latent spaces. However, we reveal that existing learnable projection mechanisms cannot ensure stable and balanced assignments from observation points to latent tokens, causing some latent tokens to be over-assigned while others remain underutilized. This limitation further restricts the design of hierarchical architectures, as assignment imbalance is continuously inherited and amplified across latent spaces, eventually causing severe token collapse in deeper spaces. To address these issues, we propose MoNo (Multiscale Optimal Transport Neural Operator), a progressive multiscale neural operator that efficiently solves PDEs on general geometries through stable latent-space construction. At its core is CoTAP (Cross-scale Optimal Transport Assignment and Projection), a novel latent-space construction method that formulates cross-space assignment between adjacent spaces as an entropy-regularized optimal transport problem, thereby constructing balanced bidirectional projections and stable latent spaces. CoTAP also ensures stable information transfer across multiple latent spaces, further enabling multiscale architectures on general geometries, which in turn support more efficient learning of long-range physical interactions. Extensive experiments demonstrate that MoNo outperforms existing state-of-the-art neural operators in both prediction performance and computational efficiency. Code is available at https://github.com/ZijiangY1116/MoNo.