MoNo：用于求解一般几何上偏微分方程的多尺度最优传输神经算子
MoNo: Multiscale Optimal Transport Neural Operator for Solving PDEs on General Geometries
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中文总结 AI 辅助
MoNo是一种多尺度最优传输神经算子，通过提出CoTAP解决了现有神经算子的令牌分配失衡与崩溃问题，在一般几何的PDE求解上，其预测性能与计算效率均优于现有最先进模型。
中文摘要 AI 辅助
基于Transformer的神经算子通过将空间观测投影为紧凑的潜在令牌并在潜在空间中学习物理交互，在求解偏微分方程（PDE）方面取得了显著进展。但我们发现，现有可学习投影机制无法确保观测点到潜在令牌的稳定均衡分配，导致部分潜在令牌被过度分配而其余令牌未被充分利用。这一限制进一步制约了分层架构的设计，因为分配失衡会在潜在空间中持续继承并放大，最终在深层空间引发严重的令牌崩溃。为解决这些问题，我们提出MoNo（Multiscale Optimal Transport Neural Operator，多尺度最优传输神经算子），这是一种渐进式多尺度神经算子，可通过稳定的潜在空间构建高效求解一般几何上的PDE。其核心是CoTAP（Cross-scale Optimal Transport Assignment and Projection，跨尺度最优传输分配与投影），一种新型潜在空间构建方法，该方法将相邻空间间的跨空间分配表述为熵正则化最优传输问题，从而构建均衡的双向投影与稳定的潜在空间。CoTAP还确保了跨多个潜在空间的稳定信息传递，进一步支持一般几何上的多尺度架构，进而实现更高效的长程物理交互学习。大量实验表明，MoNo在预测性能和计算效率上均优于现有最先进的神经算子。代码可在该https网址获取。
英文摘要
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.
