Standard data-driven models can learn physically implausible relationships, extrapolate poorly beyond their training domain, and require large datasets to achieve acceptable accuracy. Physics-informed machine learning addresses all three limitations by embedding known governing equations, conservation laws, and symmetry constraints directly into the learning framework — either through the loss function, the network architecture, or both.

This workshop surveys the landscape of physics-informed approaches, beginning with physics-informed neural networks (PINNs), which incorporate the residuals of partial differential equations as additional loss terms via automatic differentiation. It then examines operator learning methods — DeepONet and neural operators — that learn mappings between function spaces and enable rapid solution of parametric PDEs. The treatment of symmetry as inductive bias, developed through the Erlangen Programme perspective of geometric deep learning, connects to the equivariant architectures encountered in Workshops 5 and 6. Symbolic regression methods (SINDy, PySR), which aim to recover closed-form analytic equations from data, are presented as a complementary route to interpretable physical models.

Examples are drawn from fluid mechanics, quantum mechanics, reaction-diffusion systems, and electrochemical transport, with particular attention to the combination of physics constraints with experimental electrochemical data. The workshop also addresses the limitations of physics-informed approaches, examining where they succeed and where they struggle, to support realistic expectations and informed method selection.



Extended Key Takeaways & Outputs

• Understanding of PINNs, operator learning, and symbolic regression as complementary physics-informed strategies

• Practical knowledge of incorporating PDE residuals, conservation laws, and symmetry into ML frameworks

• Appreciation of the Erlangen Programme perspective on geometric deep learning

• Worked examples from fluid mechanics, quantum mechanics, and electrochemical transport

• Realistic assessment of where physics-informed methods add value and where they face limitations