Differentiable simulation

Learning goals¶

• Explain how a gradient with respect to $(H, D)$ can flow through a numerical solver

• Compare differentiable simulation with simulation-based least squares and inverse PINNs on the same data contract

• State what the method does not fix: identifiability, model mismatch, and bound activity

Reading¶

• Classical estimation, automatic differentiation, inverse PINN

• Thuerey et al. (2021), differentiable physics and differentiable physics versus physics-informed training (method background with fluid examples)

The idea¶

Simulation-based least squares already puts the simulator in the loop: pick $(H, D)$, simulate, compare with measurements. What it lacks is a cheap direction to move $(H, D)$. A grid search simulates many candidates, and finite differences need extra simulations per parameter.

Differentiable simulation writes the integrator itself in an automatic-differentiation framework. Each solver step is an ordinary differentiable operation, so reverse-mode AD returns

from a single forward rollout and one backward pass, where $\hat{\omega}_k(\theta)$ is the simulated speed at sample $k$.

How the gradient flows¶

For the classical SMIB of SMIB dynamics, one RK4 step maps the state $x_k = (\delta_k, \omega_k)$ to

where $X_k$ is the network reactance for the current stage (prefault, fault-on, postfault). The trajectory is a chain $x_0 \to x_1 \to \cdots \to x_N$, and the loss gradient is the chain rule along that chain. Backpropagation computes it automatically, carrying the sensitivity of each state to $H$ and $D$ backward through every step.

Two practical conditions make this work in the teaching model:

1. Fault switching depends on time only. The reactance changes at fixed $t_f$ and $t_c$, so put both on the step grid. Each step then sits inside one stage, and nothing non-differentiable happens inside a step. State-triggered events (protection acting on $\delta$) need event-aware solvers and are beyond this course.

2. Scales are comparable. Fit $\log H$ and $\log D$, and compare speed deviation $\omega - \omega_0$, as in gradient scaling.

Course helper: sciml_ps.smib_torch.rollout_smib (fixed-step RK4 in PyTorch, tensors (batch, time, 2)). parity_error checks it against the SciPy reference solver used in Lab 1.

Three estimators, one data contract¶

The comparison rule from Lab 3 still holds: same measurements, noise, window, and bounds for every estimator, and a held-out disturbance for every estimate.

What it does not fix¶

• Identifiability. Exact gradients of a flat loss surface are still flat. Weak excitation leaves $D$ poorly informed whichever estimator you use (identifiability).

• Model mismatch. The fitted $\theta$ is the best value for the model you differentiated. If the data come from a different model, $(\hat H, \hat D)$ become equivalent parameters (physical vs equivalent).

• Bounds. Report the bounds and whether an estimate ended on one.

Where it leads¶

Differentiable simulators are also the basis of hybrid models: a neural term trained inside the solver, next to the known swing equation. See Neural ODE / hybrid. That is how SciML models can later plug into established simulation workflows instead of replacing them.

Lab¶

• Lab 3 — Inertia and damping estimation, optional section “Differentiable simulation”

Honesty rule¶

Fast convergence is not recovery. Report the true or reference $\theta$ when known, the estimate, the bounds, and the held-out disturbance error, as for every other estimator.

Common misconceptions¶

• “Exact gradients mean the estimate is exact.”

• “Differentiable simulation is a kind of PINN.” It has no network and no residual penalty; the solver enforces the model.

• Differentiating through a state-triggered switch as if it were smooth