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Abstract Identifying the governing equations of a nonlinear dynamical system is key to both understanding the physical features of the system and constructing an accurate model of the dynamics that generalizes well beyond the available data. Achieving this kind of interpretable system identification is even more difficult for partially observed systems. We propose a machine learning framework for discovering the governing equations of a dynamical system using only partial observations, combining an encoder for state reconstruction with a sparse symbolic model. The entire architecture is trained end-to-end by matching the higher-order symbolic time derivatives of the sparse symbolic model with finite difference estimates from the data. Our tests show that this method can successfully reconstruct the full system state and identify the equations of motion governing the underlying dynamics for a variety of ordinary differential equation (ODE) and partial differential equation (PDE) systems.
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This content will become publicly available on January 1, 2026
TIME-SERIES FORECASTING AND REFINEMENT WITHIN A MULTIMODAL PDE FOUNDATION MODEL
Symbolic encoding has been used in multioperator learning (MOL) as a way to embed additional information for distinct time-series data. For spatiotemporal systems described by time-dependent partial differential equations (PDEs), the equation itself provides an additional modality to identify the system. The utilization of symbolic expressions alongside 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.
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- Award ID(s):
- 2427558
- PAR ID:
- 10632454
- Publisher / Repository:
- begellhouse
- Date Published:
- Journal Name:
- Journal of Machine Learning for Modeling and Computing
- Volume:
- 6
- Issue:
- 2
- ISSN:
- 2689-3967
- Page Range / eLocation ID:
- 77 to 89
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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