Symbolic Neural Ordinary Differential Equations
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866915191030022144 |
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| author | Li, Xin Zhao, Chengli Zhang, Xue Duan, Xiaojun |
| author_facet | Li, Xin Zhao, Chengli Zhang, Xue Duan, Xiaojun |
| contents | Differential equations are widely used to describe complex dynamical systems with evolving parameters in nature and engineering. Effectively learning a family of maps from the parameter function to the system dynamics is of great significance. In this study, we propose a novel learning framework of symbolic continuous-depth neural networks, termed Symbolic Neural Ordinary Differential Equations (SNODEs), to effectively and accurately learn the underlying dynamics of complex systems. Specifically, our learning framework comprises three stages: initially, pre-training a predefined symbolic neural network via a gradient flow matching strategy; subsequently, fine-tuning this network using Neural ODEs; and finally, constructing a general neural network to capture residuals. In this process, we apply the SNODEs framework to partial differential equation systems through Fourier analysis, achieving resolution-invariant modeling. Moreover, this framework integrates the strengths of symbolism and connectionism, boasting a universal approximation theorem while significantly enhancing interpretability and extrapolation capabilities relative to state-of-the-art baseline methods. We demonstrate this through experiments on several representative complex systems. Therefore, our framework can be further applied to a wide range of scientific problems, such as system bifurcation and control, reconstruction and forecasting, as well as the discovery of new equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_08059 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Symbolic Neural Ordinary Differential Equations Li, Xin Zhao, Chengli Zhang, Xue Duan, Xiaojun Machine Learning Mathematical Physics Differential equations are widely used to describe complex dynamical systems with evolving parameters in nature and engineering. Effectively learning a family of maps from the parameter function to the system dynamics is of great significance. In this study, we propose a novel learning framework of symbolic continuous-depth neural networks, termed Symbolic Neural Ordinary Differential Equations (SNODEs), to effectively and accurately learn the underlying dynamics of complex systems. Specifically, our learning framework comprises three stages: initially, pre-training a predefined symbolic neural network via a gradient flow matching strategy; subsequently, fine-tuning this network using Neural ODEs; and finally, constructing a general neural network to capture residuals. In this process, we apply the SNODEs framework to partial differential equation systems through Fourier analysis, achieving resolution-invariant modeling. Moreover, this framework integrates the strengths of symbolism and connectionism, boasting a universal approximation theorem while significantly enhancing interpretability and extrapolation capabilities relative to state-of-the-art baseline methods. We demonstrate this through experiments on several representative complex systems. Therefore, our framework can be further applied to a wide range of scientific problems, such as system bifurcation and control, reconstruction and forecasting, as well as the discovery of new equations. |
| title | Symbolic Neural Ordinary Differential Equations |
| topic | Machine Learning Mathematical Physics |
| url | https://arxiv.org/abs/2503.08059 |