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

TL;DR

This paper introduces MOFS, a multimodal framework enabling neural operators to generalize to unseen PDEs with minimal data, combining self-supervised pretraining, text-conditioned embeddings, and multimodal prompting.

Contribution

The work presents a novel multi-operator few-shot learning approach that integrates multimodal data and contrastive fine-tuning for PDE operator generalization.

Findings

Outperforms existing methods in few-shot PDE operator learning

Effective use of multimodal prompts improves generalization

Validated on Darcy Flow and Navier Stokes benchmarks

Abstract

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.