Automotive VHM · Operator Learning in Practice
The same operator that solves a textbook PDE can stand in for the physics of a battery pack, an inverter, a traction motor, or a suspension, fast enough to run beside live telemetry and catch faults as they grow.
Companion to the introductory monograph on FNO, KNO and DeepONet.
Recall the one line idea: a neural operator maps one whole function to another. In a vehicle, those functions are everywhere. A drive cycle is a function of time. A temperature across a pack is a function of space. A road is a function of distance. The job that turns one into another, load into heat, current into torque, road into ride, is exactly an operator.
That matters because the classical way to compute these maps, a finite element thermal model, a coupled electrochemical solver, a full vehicle dynamics simulation, is far too slow to run on a moving car. A neural operator learns the map once, offline, then evaluates it in a single fast pass. Fast enough to act as a physics surrogate living next to the sensors. And once you have a fast healthy prediction, the gap between it and the measurement, the residual, is your fault detector.
Here is the whole page in one table. Each row is a subsystem, the operator that fits it, and the failure its residual exposes.
| Subsystem | Input function | Output function | Operator that fits | What the residual reveals |
|---|---|---|---|---|
| Battery pack | load current and ambient | temperature field, state of charge | FNO for the field, KNO for aging | hot spot, capacity fade, runaway risk |
| Inverter IGBT | power loss mission profile | junction temperature trajectory | DeepONet, KNO for cycling life | rising thermal resistance from solder fatigue |
| Traction motor | operating point and rotor angle | air gap flux, torque ripple, NVH | FNO for the field, DeepONet for thermal | bearing fault, demagnetization, eccentricity |
| Suspension | road elevation profile | body motion and tire load | KNO and DeepONet for the dynamics | damper fade, bushing wear, drift in the response |
The four sections below take these one at a time, each with the plain physics, the operator mapping, a block diagram, a running simulation, and the diagnostic payoff.
Push current through a cell and it heats up: losses scale with the square of current, so a hard launch or fast charge dumps real power inside the pack. That heat spreads by diffusion and leaves through whatever cooling you have at the edges. The result is a temperature field across the pack that rises, forms a hot spot where cooling is weakest, and settles. Too hot for too long and the chemistry ages faster; far too hot and you risk thermal runaway.
The honest model is a coupled electrochemical and thermal PDE (the kind of P2D or DFN model that takes a real solver minutes per cycle). The part we can see clearly is the heat equation with internal generation:
Read it plainly: temperature changes by spreading (diffusion), plus heat made inside (generation, growing with current), minus heat carried away (cooling).
The input function is the current load over time plus the ambient condition. The output function is the whole temperature field over the pack. An FNO is a natural fit: heat diffusion is exactly the smooth, low frequency behavior its kept modes capture, and resolution invariance lets you train on a coarse pack grid and query a fine one. For the slow story, how capacity fades over thousands of cycles, a KNO marches that degradation forward to estimate remaining life.
This solves the heat equation live across a row of cells. Raise the load to dump more heat, lower the cooling to trap it, and watch a hot spot grow in the middle where cooling reaches least. The strip below is the temperature field; the curve is the same data.
The inverter switches the battery's DC into the AC the motor wants. Each switch event in the IGBT and its diode wastes a little energy (switching loss), and carrying current wastes more (conduction loss). That power turns into heat at the silicon junction, which flows down through solder layers and a baseplate into the coolant. The junction temperature Tj rises and falls with the load.
The trouble is the rise and fall itself. Every torque demand swings Tj, and because silicon, solder and copper expand by different amounts, each swing fatigues the joints. Over time solder cracks and bond wires lift. As they do, heat escapes less easily, so thermal resistance rises, and the same load now produces a hotter junction. That rising resistance is the classic degradation signature of a power module.
A common way to model the path from loss to temperature is a Foster network, a small ladder of resistor and capacitor pairs, each capturing one time scale of the heat flow:
The input function is the power loss profile over a mission; the output function is the junction temperature trajectory, or the temperature field across the die. A DeepONet maps the loss profile (branch) to Tj at any instant (trunk), replacing a slow 3D thermal model. Because the degradation accumulates slowly across many cycles, a KNO is the right tool to march that fatigue forward into a remaining life estimate.
A repeating load pulse drives the junction through thermal cycles. The teal trace is a healthy module; the rust trace is the same module with degraded solder (drag the health down). Watch the rust trace run hotter and swing wider, and watch the life estimate fall.
A permanent magnet motor makes torque from the interaction of stator currents and rotor magnets. In a perfect machine the torque would be constant, but slots, magnet edges and magnetic saturation add ripples that repeat several times per electrical revolution (the sixth order is usually the loudest). That ripple is felt as vibration and heard as whine: the NVH problem. The honest description is a magnetostatic field problem solved by finite element analysis, sweeping rotor angle and current to get the air gap flux density, and from it the torque.
One way to fight ripple, the kind of harmonic current injection used in axial flux NVH work, is to add a small current harmonic that is equal and opposite to the dominant torque harmonic, so they cancel. Get the amplitude and phase right and the ripple flattens.
The input function is the operating point and rotor angle; the output function is the air gap flux density field, and from it the torque versus angle. An FNO replaces the FEA sweep, mapping operating point to the flux field in one pass, so you can explore injection settings without re-meshing. A DeepONet maps a current trajectory to magnet temperature, which guards against demagnetization. The next demo works directly with the torque versus angle that such an operator would output.
The curve is torque across one electrical revolution, dominated by a sixth order ripple. Tune the injected harmonic's amplitude and phase to cancel it and flatten the torque. The bars show the harmonic content; the readout tracks ripple as a percentage of mean torque.
harmonic content (electrical orders)
Model a corner of the car as two masses: the body (sprung) sitting on a spring and damper, and the wheel (unsprung) on the tire's own stiffness. The road pushes the wheel, the wheel pushes the body through the suspension. The whole thing is a filter: it should pass slow road swells smoothly to keep the body calm, while keeping the tire pressed to the road for grip. Two resonances dominate, a slow body bounce near one to two hertz and a faster wheel hop near ten hertz.
The damper is the design knob, and it forces a tradeoff. Soft damping isolates the body from high frequency road but lets the body resonance run wild. Firm damping tames the resonance but transmits more harshness. This is the comfort versus control balance that active suspension, and its model based control, exists to manage.
The input function is the road elevation along the path; the output function is the body motion and tire load over time. Because the dynamics are close to linear, a KNO represents them almost exactly and marches them forward cheaply, which is ideal inside a model based controller's prediction horizon. A DeepONet maps a road profile to the ride response at any time, so you can score thousands of road realizations fast.
Left is the frequency response, how strongly each road frequency reaches the body. Right is the body's motion after hitting a bump. Drag the damping and watch the body resonance peak shrink while high frequency harshness rises, the tradeoff in one move.
body motion after a single bump
Strip away the specifics and every section ran the same loop. That loop is the reusable recipe for putting operator learning to work on a vehicle.
And the operator choice followed a simple logic each time:
A vehicle is a stack of operators: load to heat in the pack, loss to junction temperature in the inverter, current to torque in the motor, road to ride in the suspension. Each is governed by physics too slow to solve on the move, and each can be replaced by a neural operator that learns the map offline and runs in one fast pass online. FNO suits the smooth spatial fields, DeepONet suits profile to response, and KNO suits the long run forecasts. Put a surrogate beside the sensors and its residual becomes a fault detector; march it forward and it becomes a life estimate. That is operator learning earning its place in vehicle health management.