Inverse PINNs

Learning goals¶

• Formulate an inverse PINN with trainable named parameters

• Compare to classical LS on the same data contract

• Keep network inputs, known signals, and unknowns in separate buckets

Reading¶

• Inverse problems section of Raissi et al. (2019)

• Lab 3 — inertia and damping identification

Where parameters enter¶

The optimizer updates neural weights and $\theta$ (teaching Lab 3: $H$, $D$).

Buckets (do not confuse)¶

Measured power is not a parameter; it is a known signal or observation channel.

Comparison protocol (Lab 3)¶

1. Same measurements, noise model, window, and bounds

2. Classical LS baseline

3. Inverse PINN

4. Loss-surface or sensitivity diagnostic

5. Test fitted $\theta$ on a held-out disturbance

Worked example¶

Write $\mathcal{L}_{\mathrm{data}} + \lambda_{\mathrm{phys}}\mathcal{L}_{\mathrm{phys}}$ with $\theta=(H,D)$ trainable. List which tensors require requires_grad=True.

Strengths and limits for identification¶

The forward-PINN limits (strengths and limits of physics losses) carry over. Inverse use adds three of its own.

• Shared blame. Network weights and $(H, D)$ are fitted together. A flexible network can absorb model error that should have shown up in $\theta$. So a low loss does not mean $\theta$ is right.

• Weighting moves $\theta$. Changing $\lambda_{\mathrm{phys}}$ can shift the estimated $(H, D)$, not only the fit quality. Report estimates for more than one weight.

• Residual consistency is not trajectory consistency. Check the fitted $\theta$ by re-simulating with the reference solver (Lab 1). Compare that trajectory with the measurements, not the network output.

A tighter alternative puts the solver itself in the loop: differentiable simulation. There, $\theta$ is the only unknown and every candidate trajectory obeys the model exactly.

Lab¶

• Lab 3 — Inertia and damping estimation

Further reading¶

• Identifiability; identification pillar overview

• Thuerey et al. (2021), discussion of physical losses and differentiable physics versus physics-informed training

Common misconceptions¶

• Calling any PINN with parameters in the residual a solved identification problem

• Optimizing $\theta$ without bounds or units

• Using GFM triad language without declaring an elective model