{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/l-hydra-multi-head-physics-informed-neural","title":"L-HYDRA: Multi-Head Physics-Informed Neural Networks","arxiv_id":"2301.02152","date":"2023-01-05","proceeding":null,"authors":["Zongren Zou","George Em Karniadakis"],"abstract":"We introduce multi-head neural networks (MH-NNs) to physics-informed machine learning, which is a type of neural networks (NNs) with all nonlinear hidden layers as the body and multiple linear output layers as multi-head. Hence, we construct multi-head physics-informed neural networks (MH-PINNs) as a potent tool for multi-task learning (MTL), generative modeling, and few-shot learning for diverse problems in scientific machine learning (SciML). MH-PINNs connect multiple functions/tasks via a shared body as the basis functions as well as a shared distribution for the head. The former is accomplished by solving multiple tasks with MH-PINNs with each head independently corresponding to each task, while the latter by employing normalizing flows (NFs) for density estimate and generative modeling. To this end, our method is a two-stage method, and both stages can be tackled with standard deep learning tools of NNs, enabling easy implementation in practice. MH-PINNs can be used for various purposes, such as approximating stochastic processes, solving multiple tasks synergistically, providing informative prior knowledge for downstream few-shot learning tasks such as meta-learning and transfer learning, learning representative basis functions, and uncertainty quantification. We demonstrate the effectiveness of MH-PINNs in five benchmarks, investigating also the possibility of synergistic learning in regression analysis. We name the open-source code \"Lernaean Hydra\" (L-HYDRA), since this mythical creature possessed many heads for performing important multiple tasks, as in the proposed method.","url_abs":"https://arxiv.org/abs/2301.02152v1","url_pdf":"https://arxiv.org/pdf/2301.02152v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"l-hydra-multi-head-physics-informed-neural","repo_url":"https://github.com/ZongrenZou/MHPINN","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"tf","reach":{"status":"ok"}}],"tasks":[{"task_slug":"few-shot-learning","task_name":"Few-Shot Learning"},{"task_slug":"meta-learning","task_name":"Meta-Learning"},{"task_slug":"multi-task-learning","task_name":"Multi-Task Learning"},{"task_slug":"physics-informed-machine-learning","task_name":"Physics-informed machine learning"},{"task_slug":"transfer-learning","task_name":"Transfer Learning"},{"task_slug":"uncertainty-quantification","task_name":"Uncertainty Quantification"}],"methods":[{"method_slug":"normalizing-flows","method_name":"Normalizing Flows"}],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=2301.02152","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}