Scope: reduced-order modeling and surrogates
TL;DR — reduced-order modeling (ROM) and data-driven surrogates of
dynamical systems are in scope for Jaxonomy and live in
jaxonomy.library.rom. Given a full-order model (an ODE/DAE diagram or
snapshot data from one), Jaxonomy can build a cheaper reduced model that is
still a first-class, differentiable, simulatable block. What is out of
scope is the same thing as for PINNs — surrogates of spatial PDE fields
(neural operators, u(x, t) collocation); see pinn.md.
What "ROM" covers here
A reduced-order model approximates a high-dimensional or expensive dynamical system with a low-order one that is fast to simulate. Jaxonomy supports the three families that matter for control and simulation engineers, plus statistical surrogates for the input→output map:
| Family | Methods | When to reach for it |
|---|---|---|
| Linear MOR | balanced truncation (balred), minimal realization (minreal), modal truncation, singular-perturbation residualization, Krylov/IRKA (planned) |
You have (or can linearize to) an LTI model and want a smaller LTI with a certified error bound |
| Projection ROM | POD–Galerkin, Petrov–Galerkin/LSPG, DEIM hyper-reduction | You have the equations of a large nonlinear ODE/DAE (e.g. a method-of-lines PDE) and snapshots, and want an intrusive, physics-preserving reduction |
| Data-driven operator ROM | DMD, DMDc, ERA, eDMD / Koopman | You have data (snapshots), maybe no equations, and want a linear predictor — including a lifted-linear (Koopman) model you can drop straight into linear MPC/LQR |
| Statistical surrogates | Gaussian process / kriging, polynomial chaos (PCE), RBF response surface | You want a cheap, optionally uncertainty-aware surrogate of an input→output map (design maps, UQ, calibration) |
Choosing a method
- Do you have the model equations, or only data? Equations → linear MOR (if linear) or POD–Galerkin/DEIM (if nonlinear). Data only → DMD/DMDc, ERA, or eDMD/Koopman.
- Linear or nonlinear? Linear and you want a guaranteed error → balanced truncation (a priori H∞ bound). Nonlinear with equations → POD–Galerkin, and add DEIM so the per-step cost stops scaling with the full state dimension. Nonlinear with only data → eDMD/Koopman with a lifting dictionary.
- Is the reduced model for a controller? Koopman/DMDc produce a linear
reduced model in (possibly lifted) coordinates — a good basis for linear
MPC / LQR-style control (design in lifted coordinates, de-lift with
C). A lifted model wants a terminal-cost MPC rather than a hard terminal-equality one; therom_dmdc_koopman_mpcexample shows the pattern. - Is it an input→output map, not a trajectory? Use a statistical
surrogate (GP/PCE/RBF). PCE additionally yields analytic Sobol indices and
moments, so it doubles as an accelerated-UQ path into
jaxonomy.uq.
In scope — what Jaxonomy does
Every reducer returns a first-class Jaxonomy object: linear MOR returns a
reduced LinearizedSystem/LTISystem; projection and operator ROMs return a
differentiable, jit/vmap-able LeafSystem you can compose in a diagram
and drive through jaxonomy.simulate; statistical surrogates are feedthrough
blocks. ROM quality metrics (relative trajectory error, retained energy,
projection error, held-out cross-validation) live alongside the reducers.
Out of scope (use these instead)
- Spatial PDE field surrogates / neural operators (
u(x, t)over a domain, Fourier/DeepONet operators) — seepinn.md. A spatially discretized PDE (method of lines) is a large ODE and can be reduced with POD–Galerkin/DEIM here; Jaxonomy just does not own the discretization or train neural fields. - Mesh generation, CFD/FEA solvers. Bring your own high-fidelity solver; Jaxonomy reduces the resulting dynamical system or its snapshots.
The dividing line is the same as for PINNs: if the reduced object is a dynamical system evolving in time (or a map you sample), it belongs here; if it is a neural field trained against a spatial PDE residual, it does not.