Structured Simulators and Neural Operators › Choosing the Right Structure
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The four families form a design spectrum, and the choice among them is driven by what you know, what you have, and what the model is for. Reading the table by columns is instructive: the "hand-designed" column generally shrinks as you move down, while the "sweet spot" column tracks how much physics you can commit to in advance. The data-hunger labels are qualitative heuristics, not controlled cross-family measurements; they depend on the task, target error, architecture, data quality, and evaluation regime.
| Family | What is hand-designed | What is learned | Data hunger | Sweet spot |
|---|---|---|---|---|
| Differentiable simulator | Equations, integrator | Parameters, plans, residuals | Very low | Known physics, unknown constants, control |
| Graph-based models | Locality, pairwise sums, integrator | Interaction laws | Low to moderate | Many interacting objects or particles |
| Neural ODEs | Integrator, state dimension | Full vector field | Moderate | Smooth continuous-time processes, irregular data |
| Operator learning | Function-space architecture | Solution operator | Moderate to high | Fields and PDEs, multi-resolution deployment |
A few rules of thumb distilled from the chapter:
- Differentiate what you know, learn what you do not. If the equations are right up to parameters, use a differentiable simulator and fit the parameters. If only the interaction structure is known, learn inside a graph model. Reserve learning capacity for unknown components. Every valid equation you supply is one less relation the data must identify.
- Match the architecture's bias to the physics' structure. Pairwise locality calls for graphs, temporal smoothness calls for differential equations, and spatial nonlocality calls for operators. A mismatched bias can waste capacity or exclude the target dependency entirely, as the pointwise heat baseline demonstrates.
- Prefer hypothesis classes that contain the target dependency structure and can approximate the dynamics to the fidelity the task requires. The pointwise MLP failed the heat equation not for lack of width but because it had no pathway for cross-location dependence. Extra width cannot repair a missing dependency pathway.
- Hybrid physics-plus-residual models are useful when core physics is trustworthy and systematic residual effects are learnable. Physics-informed neural networks likewise add differential-equation residuals and boundary or initial-condition information to the objective, although some variants enforce conditions directly. A persistent, validated residual can flag where a mechanistic model disagrees with data, but attributing it to missing physics requires ruling out observation noise, state-estimation bias, discretization error, parameter error, distribution shift, and failed optimization.