提出新型神经算子MoNo,解决PDE求解中隐空间分配不均问题。
MoNo: Multiscale Optimal Transport Neural Operator for Solving PDEs on General Geometries

- 引入跨尺度最优传输机制,实现稳定双向投影
- 在多个数据集上预测误差降低12%-28%,计算效率提升30%以上
- 适合复杂几何上长程物理交互建模,尤其适合多尺度问题
基于Transformer的神经算子通过将空间观测投影到紧凑的隐空间令牌,并在隐空间中学习物理相互作用,在求解偏微分方程(PDEs)方面取得了显著进展。然而,我们发现现有的可学习投影机制无法保证从观测点到隐令牌的稳定均衡分配,导致部分隐令牌过度使用而其他则未被充分利用。这一限制进一步阻碍了分层架构的设计,因为分配失衡会在隐空间间持续传递并放大,最终导致深层空间出现严重令牌坍缩。为解决这些问题,我们提出了MoNo(Multiscale Optimal Transport Neural Operator),一种通过稳定隐空间构建高效求解一般几何上PDE的渐进式多尺度神经算子。其核心是CoTAP(Cross-scale Optimal Transport Assignment and Projection),一种新颖的隐空间构建方法,将相邻空间间的跨空间分配形式化为熵正则化的最优传输问题,从而实现平衡的双向投影和稳定的隐空间。CoTAP还确保了多隐空间间的信息稳定传递,进一步支持在一般几何上构建多尺度架构,从而更高效地学习长程物理相互作用。大量实验表明,MoNo在预测性能和计算效率方面均优于现有最先进神经算子。代码已开源:https://github.com/ZijiangY1116/MoNo。
原文摘要 · Abstract (English)
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.
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