MoNo: Multiscale Optimal Transport Neural Operator for Solving PDEs on General Geometries

2026-08-10

AI Summary

This work addresses the token collapse issue in existing Transformer-based neural operators when solving partial differential equations (PDEs) on general geometries, which arises from imbalanced assignments of observation points to latent tokens in deep architectures. To mitigate this, the authors propose MoNo—a progressive multiscale neural operator that, for the first time, incorporates entropy-regularized optimal transport into latent space construction. MoNo introduces a Cross-scale optimal Transport Assignment and Projection (CoTAP) mechanism, enabling balanced bidirectional token mappings and efficient cross-scale information transfer. This approach effectively alleviates token assignment imbalance, supports stable multiscale modeling on arbitrary geometries, and significantly enhances the learning of long-range physical interactions. Experiments demonstrate that MoNo consistently outperforms state-of-the-art neural operators across multiple PDE tasks, achieving superior accuracy and computational efficiency.

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.

Problem

• neural operators
• optimal transport
• PDEs
• latent token assignment
• multiscale architectures

Innovation

• Optimal Transport
• Neural Operator
• Multiscale Architecture
• PDE Solving
• Latent Space Projection