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# ConCerNet: A Contrastive Learning Based Framework for Automated Conservation Law Discovery and Trustworthy Dynamical System Prediction

*Proceedings of the 40th International Conference on Machine Learning*, PMLR 202:41694-41714, 2023.

#### Abstract

Deep neural networks (DNN) have shown great capacity of modeling a dynamical system; nevertheless, they usually do not obey physics constraints such as conservation laws. This paper proposes a new learning framework named $\textbf{ConCerNet}$ to improve the trustworthiness of the DNN based dynamics modeling to endow the invariant properties. $\textbf{ConCerNet}$ consists of two steps: (i) a contrastive learning method to automatically capture the system invariants (i.e. conservation properties) along the trajectory observations; (ii) a neural projection layer to guarantee that the learned dynamics models preserve the learned invariants. We theoretically prove the functional relationship between the learned latent representation and the unknown system invariant function. Experiments show that our method consistently outperforms the baseline neural networks in both coordinate error and conservation metrics by a large margin. With neural network based parameterization and no dependence on prior knowledge, our method can be extended to complex and large-scale dynamics by leveraging an autoencoder.