Learning goals¶
Contrast collocation PINNs with solver-based learned dynamics
State when Neural ODEs or hybrid/UDE models are appropriate teaching choices
Keep operator learning (DeepONet/FNO) as a different SciML branch
Reading¶
Neural ODEs Chen et al., 2018
Universal differential equations Rackauckas et al., 2020
Collocation PINN vs Neural ODE¶
| Collocation PINN | Neural ODE / hybrid | |
|---|---|---|
| Dynamics | Residual at collocation points | Integrate a learned or hybrid vector field with a solver |
| Strength | Flexible collocation; easy physics residual | Respects solver structure; natural rollouts |
| Watch-outs | Residual ≠ solution quality | Long-rollout drift; adjoint cost |
Hybrid / UDE idea: known physics terms plus a neural correction for missing dynamics. The correction can absorb parameter error — treat fits as possibly equivalent.
Operator learning (awareness only)¶
DeepONet / FNO learn maps between functions (e.g. load profile → trajectory). That is not the same as collocation inverse PINN for a few named scalars. See SciML landscape and Lab 4 options.
Worked example¶
Define a held-out longer time horizon before training a Neural ODE on SMIB data. Report error at train horizon vs extrapolated horizon.
Lab¶
Further reading¶
Common misconceptions¶
Treating DeepONet as “a type of PINN”
Claiming generalization without a pre-registered held-out domain
Using hybrid corrections as proof of physical parameter recovery
- Chen, R. T. Q., Rubanova, Y., Bettencourt, J., & Duvenaud, D. (2018). Neural Ordinary Differential Equations. arXiv. 10.48550/arXiv.1806.07366
- Rackauckas, C., Ma, Y., Martensen, J., Warner, C., Zubov, K., Supekar, R., Skinner, D., Ramadhan, A., & Edelman, A. (2020). Universal Differential Equations for Scientific Machine Learning. arXiv. 10.48550/arXiv.2001.04385