Interactive Monograph · Scientific ML for Automotive
A companion to the first monograph, now under the hood. The same idea, a network taught by data and by governing equations at once, applied to the three subsystems that decide an electric drivetrain's range, reliability, and torque: the battery, the inverter, and the motor.
A car is instrumented sparsely on purpose. Every sensor is cost, weight, and another thing that can fail, so the quantities engineers most want, the charge left in a pack, the temperature inside a power module, the flux in a rotor, are usually not measured directly. They are inferred from a few cheap signals and a model of how the part behaves.
That is already the logic of a state observer, the Kalman filters and Luenberger observers that run in every modern controller. A Physics-Informed Neural Network is a close cousin: instead of hand-deriving an observer, you let a network learn the mapping while a governing equation keeps it honest. The payoff for automotive work is specific:
The three sections that follow each take a real drivetrain subsystem, state its governing equation, simulate it live, and show where a PINN attaches. The simulations below are genuine numerical integrations of those equations, running in your browser.
Every example uses the same skeleton from the first monograph. A network takes the operating variables and predicts a state. That prediction is graded two ways: against whatever was measured, and against the subsystem's differential equation, evaluated by automatic differentiation at many points. The only thing that changes from battery to inverter to motor is which equation goes in the physics box.
A lithium cell is modeled as a controlled voltage source behind a small network of resistors and capacitors. The source is the open-circuit voltage, a known function of how full the cell is. A series resistor R₀ captures the instant voltage drop under load, and a resistor-capacitor pair, R₁ with C₁, captures the slower polarization that makes voltage sag and recover. This is the Thevenin equivalent-circuit model that runs in most battery management systems.
Run a load profile through the cell. The amplitude slider scales the current draw; the other two age the cell, R₀ rises and capacity Q fades as a battery wears. Watch the terminal voltage sag harder and the usable charge shrink.
For state estimation, the network takes time and the measured current and predicts SoC and V₁. The data loss matches the one signal you actually have, terminal voltage; the physics loss enforces the two differential relations at many instants. The network learns to back out charge from voltage exactly as an observer would, with the circuit equations as the law.
Turn it around for diagnostics. If R₀ is unknown, make it trainable and recover it from operating data. A rising R₀ is a direct state-of-health marker. Press estimate to fit R₀ to the noisy voltage above; the aging gauge reads the deviation from a fresh cell.
The inverter switches the DC bus into three-phase current, and every switching event and every amp of conduction dumps heat into the power module. The silicon junction is the hottest, most fragile point in the drivetrain, and it is never measured directly, only a slower case or coolant sensor is. Junction temperature must be estimated, and it is what limits how hard the inverter can be pushed.
Two physics blocks describe it. A loss model turns current and switching frequency into power dissipated, and a thermal network, a ladder of thermal resistances and capacitances, turns that power into a temperature rise over time.
Drive a repeating load, like bursts of hard acceleration. The junction heats and cools, the swing being the thermal cycling that fatigues solder and bond wires. The cooling slider degrades the heat path, as a dried-out thermal interface would, and the limit line marks where the controller must derate.
The network maps time and load to junction temperature, with the data loss tied to the slow case sensor you can afford and the physics loss enforcing the thermal network. It becomes a soft sensor for the temperature you cannot measure. Make the thermal resistance trainable and a rising value reveals a degrading heat path before the junction ever overheats, the prognostic version of the same model.
The traction motor is governed cleanest in the rotating dq frame, where the three phase currents become two: id aligned with the rotor magnet and iq in quadrature, the one that makes torque. In that frame the electrical dynamics are two coupled first-order equations, and the torque is an algebraic combination of the currents.
Two features make this rich. The red cross-coupling terms mean the d and q axes disturb each other more violently the faster the motor spins, so a controller must actively cancel them. And the saliency term, present when Ld and Lq differ, is the extra reluctance torque your axial-flux and interior-magnet machines are designed to exploit.
Command a step in iq and watch the loop respond. With decoupling off, spinning fast, the q-axis step kicks the d-axis current sideways through the cross terms. Switch decoupling on and the disturbance is cancelled, the clean axis separation that field-oriented control exists to deliver.
Feed the network voltages and speed and have it predict the dq currents, with the two electrical equations as the physics. More valuable on a wearing motor: make the parameters trainable. A drifting Rₛ flags a winding or connection fault, changing Ld and Lq reveal magnetic saturation, and a falling ψₘₐ is demagnetization, a partially lost magnet. Estimate the flux from the steady operating point below.
Each subsystem carried its own equation, but they share one diagnostic principle: train a network to obey the healthy physics, then watch the residual. A residual that stays near zero means the part still behaves lawfully; a residual that creeps up is a part beginning to fail. Run one such monitor per subsystem and feed the residuals to a supervisor and you have the skeleton of a model-based health manager.
A PINN does not replace the extended Kalman filters, sliding-mode observers, and calibrated lookup tables already in production. It complements them. Where a hand-built observer needs a tractable model and careful tuning, a PINN can absorb messier physics and learn corrections from fleet data, while the equation keeps it from hallucinating. The honest framing is a hybrid model: physics for the part you trust, learning for the part you cannot derive.
Used with these in mind, physics-informed learning is a natural fit for the parts of a vehicle that are hard to sense and expensive to fail: a way to turn the equations engineers already trust into soft sensors and health monitors.