Symbolic encoding has been used in multi-operator learning as a way to embed additional information for distinct time-series data. For spatiotemporal systems described by time-dependent partial differential equations, the equation itself provides an additional modality to identify the system. The utilization of symbolic expressions along side time-series samples allows for the development of multimodal predictive neural networks. A key challenge with current approaches is that the symbolic information, i.e. the equations, must be manually preprocessed (simplified, rearranged, etc.) to match and relate to the existing token library, which increases costs and reduces flexibility, especially when dealing with new differential equations. We propose a new token library based on SymPy to encode differential equations as an additional modality for time-series models. The proposed approach incurs minimal cost, is automated, and maintains high prediction accuracy for forecasting tasks. Additionally, we include a Bayesian filtering module that connects the different modalities to refine the learned equation. This improves the accuracy of the learned symbolic representation and the predicted time-series.

本研究解决了现有时间序列预测中符号信息预处理的复杂性问题，提出基于SymPy的新型token库，将微分方程编码为时间序列模型的额外模态。该方法自动化程度高、成本低，且在预测任务中保持了高精度，显著提高了学习到的符号表示和预测时间序列的准确性。

多模态偏微分方程基础模型中的时间序列预测、知识蒸馏与精细化