arXiv:2508.01211cs.LG2025-08被引 1

用少量例子让AI学会跨方程族的解算,突破传统方法依赖大量数据的瓶颈。

Multi-Operator Few-Shot Learning for Generalization Across PDE Families

  • 通过多任务自监督预训练共享编码器,学习空间场重建与频谱预测。
  • 在少样本条件下对未见方程族实现更优解算性能,显著超越现有基线。
  • 融合视觉、频域与文本模态,适合需要快速适应新物理场景的研究者。

求解偏微分方程(PDE)的解算算子已成为科学机器学习的基础任务。然而,现有神经算子方法对每类特定PDE需大量训练数据,且难以跨方程族泛化。本文提出MOFS:一种统一的多算子少样本学习框架,旨在仅用少量示范样例即可泛化至未见的PDE算子。方法包含三部分:(i) 共享傅里叶神经算子(FNO)编码器的多任务自监督预训练,用于重构掩码空间场并预测频谱;(ii) 基于输入输出场统计摘要生成的文本条件算子嵌入;(iii) 带门控融合与跨模态梯度注意力的记忆增强多模态提示。采用两阶段训练:先在已见算子上学习提示条件推理,再通过端到端对比微调对齐视觉、频域与文本模态的隐表示。在Darcy Flow与Navier-Stokes变体等PDE基准测试中,模型在少样本泛化上优于现有算子学习基线。大量消融实验验证了各模态与训练组件的贡献。该方法为跨科学领域通用且数据高效的算子学习提供了新基础。

原文摘要 · Abstract (English)

Learning solution operators for partial differential equations (PDEs) has become a foundational task in scientific machine learning. However, existing neural operator methods require abundant training data for each specific PDE and lack the ability to generalize across PDE families. In this work, we propose MOFS: a unified multimodal framework for multi-operator few-shot learning, which aims to generalize to unseen PDE operators using only a few demonstration examples. Our method integrates three key components: (i) multi-task self-supervised pretraining of a shared Fourier Neural Operator (FNO) encoder to reconstruct masked spatial fields and predict frequency spectra, (ii) text-conditioned operator embeddings derived from statistical summaries of input-output fields, and (iii) memory-augmented multimodal prompting with gated fusion and cross-modal gradient-based attention. We adopt a two-stage training paradigm that first learns prompt-conditioned inference on seen operators and then applies end-to-end contrastive fine-tuning to align latent representations across vision, frequency, and text modalities. Experiments on PDE benchmarks, including Darcy Flow and Navier Stokes variants, demonstrate that our model outperforms existing operator learning baselines in few-shot generalization. Extensive ablations validate the contributions of each modality and training component. Our approach offers a new foundation for universal and data-efficient operator learning across scientific domains.

PDE求解少样本学习多模态神经算子

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