Learning and Transferring Physical Models through Derivatives

We propose Derivative Learning (DERL), a supervised approach that models physical systems by learning their partial derivatives. We also leverage DERL to build physical models incrementally, by designing a distillation protocol that effectively transfers knowledge from a pre-trained to a student model. We provide theoretical guarantees that our approach can learn the true physical system, being consistent with the underlying physical laws, even when using empirical derivatives. DERL outperforms state-of-the-art methods in generalizing an ODE to unseen initial conditions and a parametric PDE to unseen parameters. We finally propose a method based on DERL to transfer physical knowledge across models by extending them to new portions of the physical domain and new range of PDE parameters. We believe this is the first attempt at building physical models incrementally in multiple stages.

@article{trenta2025_2505.01391,
  title={ Learning and Transferring Physical Models through Derivatives },
  author={ Alessandro Trenta and Andrea Cossu and Davide Bacciu },
  journal={arXiv preprint arXiv:2505.01391},
  year={ 2025 }
}