Forward PINNs

Learning goals¶

Write data and physics losses for a known ODE fragment
Explain the role of the physics weight $\lambda_{\mathrm{phys}}$
Contrast forward PINNs with supervised surrogates

Reading¶

Original PINN formulation Raissi et al., 2019

Three pieces¶

Neural network — usually input $t$ (and maybe known parameters / controls); output $\hat{x}(t)$
Data loss — match measurements or labeled simulator trajectories

\mathcal{L}_{\mathrm{data}} \approx \|\hat{x}(t) - x^{\mathrm{meas}}(t)\|^2

(1)

Physics loss — collocation residual of the ODE/DAE fragment

\mathcal{L}_{\mathrm{phys}} \approx \|r(\hat{x},\dot{\hat{x}},u,\theta)\|^2

(2)

Total loss:

\mathcal{L} = \mathcal{L}_{\mathrm{data}} + \lambda_{\mathrm{phys}}\mathcal{L}_{\mathrm{phys}} + \cdots

(3)

Plain words: the network is pulled toward the data and toward obeying physics. $\lambda_{\mathrm{phys}}$ sets how hard the physics pull is.

Forward-style use¶

In a forward PINN, parameters $\theta$ inside the residual are known and fixed. The network learns trajectories (or a surrogate solution) consistent with physics and any available labels.

If $\lambda_{\mathrm{phys}}=0$	If you ignore data
You may fit waveforms that violate physics	You may get a residual-minimizing curve that misses measurements

Worked example¶

For the classical swing equations with known $(H,D)$ , place collocation points in time and write $r_\delta$ and $r_\omega$ explicitly. Train with a small labeled window plus residual.

Strengths and limits of physics losses¶

The points below follow the discussion of physical losses in Thuerey et al. (2021), rewritten for the swing equation.

Strength	Limit
Needs no solver: only the network, AD, and the residual $r_\delta, r_\omega$	Soft constraint. The residual is only penalized, so a trained $\hat{x}(t)$ still violates the swing equation by some amount. Report the residual next to trajectory error
Works with sparse or partial measurements (e.g. only $\omega$ measured)	Collocation cost. The residual is only checked at collocation points. Fault switching at $t_f$ and $t_c$ needs points concentrated around the events
Same code pattern for forward and inverse problems	Loss weighting. $\mathcal{L}_{\mathrm{data}}$ , $r_\delta$ and $r_\omega$ have different scales, so $\lambda_{\mathrm{phys}}$ (and per-term weights) change the answer. See gradient scaling
Smooth, differentiable output in $t$	Weak extrapolation. A network trained on $t \in [0, 2]$ s is not a simulator. Outside the training window, or for a new clearing time, it must be re-checked against Lab 1

Plain words: a PINN is a curve fitted to data and nudged toward the physics. It is not a solver that guarantees the physics.

Lab¶

Lab 2 — Surrogate versus PINN
Optional further practice (does not replace Lab 2): ANDES PINN surrogate gallery

Common misconceptions¶

Calling every residual network an “inverse PINN”
Reporting only $\mathcal{L}_{\mathrm{phys}}$ without trajectory metrics
Assuming a trained forward PINN is a stability certificate

References¶

Raissi, M., Perdikaris, P., & Karniadakis, G. E. (2019). Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics, 378, 686–707. 10.1016/j.jcp.2018.10.045
Thuerey, N., Holzschuh, B., Holl, P., Kohl, G., Lino, M., Liu, Q., Schnell, P., & Trost, F. (2021). Physics-based Deep Learning. arXiv. 10.48550/arXiv.2109.05237

Learning goals¶

Reading¶

Three pieces¶

Forward-style use¶

Worked example¶

Strengths and limits of physics losses¶

Lab¶

Further reading¶

Common misconceptions¶