Source-linked AI summary
Operator inference for non-intrusive model reduction of systems with non-polynomial nonlinear terms
Peter Benner, Pawan Goyal, Boris Kramer, Benjamin Peherstorfer, Karen Willcox
TL;DR
Non-intrusive nonlinear model reduction is comparatively scarce, while intrusive methods require full equation and discretization knowledge. This work learns reduced operators from snapshots while incorporating analytically specified, spatially local non-polynomial terms; experiments demonstrate predictive reduced models and the paper proves convergence to intrusive reduced models under stated conditions.
Problem
Intrusive reduction requires full governing-equation and discretization knowledge, while non-intrusive methods for nonlinear reduced models are comparatively scarce.
Source: §1 · p. 1Method
Operator inference learns reduced operators from snapshots while incorporating known spatially local non-polynomial terms, with empirical interpolation supporting online efficiency.
Source: §1 · p. 2, §2.4 · p. 4, §3.1 · p. 5Results
Under mild assumptions and sufficiently small sampling time steps, learned reduced models converge to intrusive reduced models; three experiments demonstrate performance on strongly nonlinear systems and chemical processes.
Source: §5 · pp. 15–16Takeaways & Limitations
The experiments show that known structure in strongly nonlinear systems can be embedded in the learning framework, which produces predictive reduced models for diffusion-reaction and chemical-process problems.
Source: §1 · p. 2, §5 · p. 16The method considers continuous, spatially local nonlinear terms and excludes terms involving spatial differentials or integrals of the state.
Source: §2.2 · p. 3
Abstract
from arXiv · showhide
This work presents a non-intrusive model reduction method to learn low-dimensional models of dynamical systems with non-polynomial nonlinear terms that are spatially local and that are given in analytic form. In contrast to state-of-the-art model reduction methods that are intrusive and thus require full knowledge of the governing equations and the operators of a full model of the discretized dynamical system, the proposed approach requires only the non-polynomial terms in analytic form and learns the rest of the dynamics from snapshots computed with a potentially black-box full-model solver. The proposed method learns operators for the linear and polynomially nonlinear dynamics via a least-squares problem, where the given non-polynomial terms are incorporated in the right-hand side. The least-squares problem is linear and thus can be solved efficiently in practice. The proposed method is demonstrated on three problems governed by partial differential equations, namely the diffusion-reaction Chafee-Infante model, a tubular reactor model for reactive flows, and a batch-chromatography model that describes a chemical separation process. The numerical results provide evidence that the proposed approach learns reduced models that achieve comparable accuracy as models constructed with state-of-the-art intrusive model reduction methods that require full knowledge of the governing equations.
1. Introduction
Traditional model reduction often requires access to full governing equations and discretizations, limiting its use when those details are unavailable. This work learns reduced operators from snapshots while incorporating analytically known, spatially local non-polynomial terms.
1. Introduction
- Traditional intrusive techniques require full knowledge of governing equations and discretizations, while nonlinear reduced-order models learned non-intrusively remain comparatively scarce.This limits intrusive methods when, for example, proprietary simulation software does not expose its equations, discretization, or solver.Source: §1 · p. 1
- Existing operator inference applies to polynomial nonlinearities, while extensions to general nonlinear systems include variable transformations.Other non-intrusive nonlinear approaches use bilinear or quadratic-bilinear parametrizations, selected function libraries, or suitable observables.Source: §1 · p. 2
- The method incorporates analytically known non-polynomial terms and learns the remaining reduced operators from snapshots, converging to intrusive reduced models under certain conditions.The approach combines operator inference with empirical interpolation for online efficiency and can retain structure in coupled PDEs.Source: §1 · p. 2
2. Problem setting and preliminaries
The section specifies a PDE and its semi-discrete model, reviews projection-based reduction, and defines the data-access setting for learning a nonlinear reduced model. The target setting has snapshot data and a spatially local nonlinear term, but unavailable discretized operators.
2.1. Partial differential equation model
- The PDE setting assumes given initial and boundary conditions that make the initial-boundary value problem well posed, with linear, quadratic, nonlinear, and input terms.The model allows spatial domains in one, two, or three dimensions.Source: §2.1 · pp. 2–3
2.2. Discretized model
- The semi-discrete full-order model has finite-dimensional states and operators, and the considered non-polynomial terms are spatially local and evaluated componentwise.Functions involving spatial differentials or integrals of the state are outside the considered class.Source: §2.2 · p. 3
2.3. Traditional projection-based model reduction for dynamical systems
- Projection-based reduction uses a POD basis computed from snapshots and restricts the full-order operators to that subspace, requiring access to those operators.The reduced nonlinear term still requires evaluations of the original function; duplicate quadratic terms can be removed from its reduced representation.Source: §2.3 · p. 4
2.4. Problem setting
- The learning problem assumes state and input snapshots are available while the discretized operators are not, and seeks a nonlinear reduced-order model.The available nonlinear term is spatially local.Source: §2.4 · p. 4
3. Operator inference for nonlinear systems
Operator inference learns reduced operators from snapshots while retaining analytically available spatially local nonlinear terms. Under time-discretization and rank conditions, the learned operators approach their intrusive projection counterparts; empirical interpolation can reduce nonlinear-term evaluation cost.
3.1. Operator inference method
- The method extends operator inference to spatially local non-polynomial terms, using analytic evaluations of those terms while fitting the remaining reduced operators by least squares.Snapshots can be post-processed to evaluate the known nonlinear term, which is incorporated into the least-squares problem's known terms.Source: §3.1 · p. 5
3.2. Analysis of operator inference framework
- Under the stated time-discretization assumptions and a full-column-rank data matrix, the learned operators can approximate the intrusive projected operators within any ε > 0.The theorem gives Frobenius-norm error bounds for the learned linear, input, and quadratic operators.Source: §3.2 · p. 7
- The convergence result has no rate, and inferred-operator errors may not decrease monotonically at low ROM orders.For the tubular reactor, the relative error in inferred linear operators is 1.66 × 10^-2 even at r = 50; the reported numerical results nevertheless show accurate learned ROMs with convergent state errors for r ≪ n.Source: §3.2 · p. 7
- When k < n, operator inference and projection-based POD have similar costs, with both including operations that scale as O(rn).Operator inference additionally forms the reduced quadratic-state data and solves a reduced-dimensional least-squares problem; projection may incur expense forming V⊤H(V ⊗′ V).Source: §3.2 · p. 7
3.3. Empirical interpolation
- DEIM approximates the nonlinear-term evaluation using a basis from the leading left singular vectors of nonlinear snapshots, evaluating only m ≪ n entries of f for speedup.The selection operator enforces interpolation and limits the local growth of ∥(S⊤W)^−1∥2, which appears in the DEIM error bound.Source: §3.3 · p. 8
4. Numerical results
The numerical examples compare learned ROMs with intrusive POD ROMs across three PDE models, using snapshots and known local nonlinear terms to build reduced models. Across the examples, the learned models achieve comparable accuracy, with performance depending on model order and preservation of coupling structure.
4. Numerical results
- The study tests the approach on three examples of increasing complexity and compares learned ROMs with projection-based POD ROMs.The examples are the Chafee-Infante diffusion-reaction model, a tubular reactor, and batch chromatography.Source: §4 · p. 8, §4.1 · p. 8, §4.2 · p. 9
4.1. Chafee-Infante model
- In the Chafee-Infante example, the learned ROM is more accurate at most output time steps and slightly more accurate in state approximation than POD, but becomes unstable after r = 14.The authors suggest small higher-mode singular values and limited training information may contribute; output accuracy is already 10^-7 at that point.Source: §4.1 · p. 9
4.2. Tubular reactor model
- In the tubular reactor, learned and projection-based POD ROMs have similar output errors through a simulation 100% longer than training, while learned-ROM state errors are lower in most cases.State errors for both models level off at 10^-7 after r = 18.Source: §4.2 · p. 11
4.3.1. Problem setup
- Batch chromatography couples four concentration and solid-phase fields, and its rational adsorption nonlinearity is spatially local, placing it within the proposed framework.The process separates mixture components because they have different adsorption affinities and move at different velocities.Source: §4.3.1 · pp. 11–13
4.3.2. Maintaining the coupling structure: Projection-based ROM
- For batch chromatography, conventional shared-subspace POD produced unstable ROMs; preserving subsystem coupling yielded stable ROMs, though a theoretical justification for nonlinear systems remains open.The proposed projection uses separate bases for each concentration and solid-phase state.Source: §4.3.2 · p. 14
4.3.3. Maintaining the coupling structure: Learned ROM
- The learned batch-chromatography ROM enforces coupling by using a block-diagonal projection and solving separate least-squares problems for the coupled system components.No least-squares problem is needed for the solid-phase derivative equations because their right-hand sides have no linear terms.Source: §4.3.3 · pp. 14–15
4.3.4. Results
- For r < 30, batch-chromatography intrusive and learned ROMs perform similarly, but learned-ROM state errors decrease less favorably at higher orders.The authors suggest operator-inference conditioning may contribute; regularization or data from multiple simulations could improve performance.Source: §4.3.4 · p. 15
5. Conclusions
The method learns reduced models non-intrusively for spatially local non-polynomial systems, and the experiments show accurate predictions for strongly nonlinear systems. Under stated assumptions, learned ROM operators converge to the projected operators as the ROM order approaches the full dimension.
5. Conclusions
- The method uses full-model state data to learn reduced models for spatially local non-polynomial nonlinearities, and exploits locality to generate nonlinear snapshots without explicit access to the nonlinear-term discretization.Under mild time-stepping assumptions and sufficiently small sampling intervals, the learned operators converge to the projected ROM operators as r approaches n.Source: §5 · p. 15
- Figure 10 compares the intrusive POD and learned models at order r = 4×22, while Figure 11 shows their relative errors for batch chromatography.The captions identify the comparisons but do not state which model performs better.Source: Figure 10 · p. 16, Figure 11 · p. 16
- The three numerical experiments show the method works well for strongly nonlinear systems and can embed additional known structure in the learning framework.The authors identify measurement noise, regularization, and physical constraints as directions for future work.Source: §5 · p. 16
- Future work includes studying measurement noise, developing regularization to improve accuracy and stability, and incorporating physical constraints into the least-squares problem.The physical-constraints direction is suggested by.Source: §5 · p. 16