Carleman Linearization Overview

• Carleman linearization is a method that lifts nonlinear systems into an infinite-dimensional linear framework using monomials, Fourier modes, or shifted coordinates to enable systematic analysis.
• Finite-section approximations truncate the infinite hierarchy, with convergence theory providing explicit error bounds under stability and spectral conditions.
• The approach supports various applications—from nonlinear control to quantum simulation—by transforming complex dynamics into more tractable linear representations.

Carleman linearization, also called Carleman embedding, is a procedure that replaces a nonlinear dynamical system by an infinite-dimensional linear system built from lifted observables of the original state. In the classical formulation, the lift consists of monomials or Kronecker powers of the state; in more recent formulations, the same embedding logic is adapted to Fourier modes, tensor powers on Hilbert spaces, shifted coordinates, and piecewise local charts. The method is exact only at the level of the infinite hierarchy. Practical use therefore centers on finite truncations, their error bounds, and the structural conditions under which the first lifted block approximates the original nonlinear trajectory with controlled accuracy (Amini et al., 2022, Motee et al., 3 Mar 2025).

1. Classical monomial embedding

For polynomial or analytic ODEs near an equilibrium, Carleman linearization starts from a representation such as

$x'(t)=F_1x+F_2x^{[2]}+\cdots+F_kx^{[k]},$

or, in scalar form,

$\dot x=g(x)=\sum_{m\ge 0} c_m x^m.$

The state is lifted to monomials or Kronecker powers,

$y_i:=x^{[i]},\qquad i\in\mathbb N,$

so that differentiation and Leibniz’ rule yield a linear hierarchy

$y_i'(t)=\sum_{j=0}^{k-1} A^i_{i+j} y_{i+j}(t).$

Stacking the blocks produces an infinite block upper-triangular linear system. In the scalar Maclaurin setting, the lifted vector is

$\mathbf{x}=[x,x^2,x^3,\dots]^\mathsf T,$

and the resulting infinite matrix is built directly from the coefficients $c_m$ (Forets et al., 2017, Motee et al., 3 Mar 2025).

The structural reason the method is useful is that degree is preserved by the linear part and raised in a controlled way by the nonlinear terms. In the homogeneous case $f(0)=0$, the lifted operator is block upper triangular, and the original state appears in the first block. A truncation at order $N$ keeps only finitely many lifted coordinates and defines the finite-section approximation. In the analytic ODE framework of (Amini et al., 2022), the truncated system has dimension

$\binom{N+d}{d}-1,$

while in quadratic reachability formulations the lifted dimension scales as

$n+n^2+\cdots+n^N=\frac{n^{N+1}-n}{n-1}.$

This tradeoff—linearity in lifted space versus rapid growth of lifted dimension—is the defining computational feature of the method (Amini et al., 2022, Forets et al., 2021).

A common misconception is that Carleman linearization yields a finite exact linear model. The exact equivalence is with the infinite hierarchy; truncation is the approximation step. The resulting linear surrogate can be highly effective, but its validity depends on truncation order, time horizon, and the analytic or spectral assumptions used in the convergence theory (Forets et al., 2017, Amini et al., 2022).

2. Finite-section approximations and convergence theory

A central problem is to determine when the first block of a finite truncation converges to the true nonlinear trajectory as the truncation order increases. Early explicit results for polynomial ODEs provided local, time-dependent truncation-error bounds. For quadratic systems, one bound in (Forets et al., 2017) uses an a priori estimate $\dot x=g(x)=\sum_{m\ge 0} c_m x^m.$0, while a second, generating-function-based bound depends only on $\dot x=g(x)=\sum_{m\ge 0} c_m x^m.$1, $\dot x=g(x)=\sum_{m\ge 0} c_m x^m.$2, and $\dot x=g(x)=\sum_{m\ge 0} c_m x^m.$3. The paper emphasizes that the second strategy is effectively computable and yields a closed-form convergence radius $\dot x=g(x)=\sum_{m\ge 0} c_m x^m.$4 (Forets et al., 2017).

For analytic time-varying systems, (Amini et al., 2022) establishes explicit finite-section bounds under a uniform exponential decay assumption on the Maclaurin coefficients. If

$\dot x=g(x)=\sum_{m\ge 0} c_m x^m.$5

then the first block of the finite-section approximation converges exponentially in truncation order on an explicit finite interval $\dot x=g(x)=\sum_{m\ge 0} c_m x^m.$6. Under an additional Hurwitz-type Jacobian condition at the origin, the same paper proves exponential convergence uniformly for all $\dot x=g(x)=\sum_{m\ge 0} c_m x^m.$7. This whole-time result is one of the main distinctions between merely local convergence theory and asymptotic stability-based theory (Amini et al., 2022).

Subsequent work weakens the classical dissipativity assumptions. In the quantum-simulation setting of (Wu et al., 2024), the system

$\dot x=g(x)=\sum_{m\ge 0} c_m x^m.$8

is analyzed under spectral non-expansion and a no-resonance condition

$\dot x=g(x)=\sum_{m\ge 0} c_m x^m.$9

rather than strict dissipation. The resulting estimate

$y_i:=x^{[i]},\qquad i\in\mathbb N,$0

shows convergence controlled by the resonance gap $y_i:=x^{[i]},\qquad i\in\mathbb N,$1. This broadens the regime from strongly dissipative systems to oscillatory and dispersive examples such as KdV and FPU chains (Wu et al., 2024).

A further refinement appears in the observable-specific analysis of (Boreale et al., 23 Jun 2026). Instead of bounding the entire retained tail uniformly, that paper works directly in the original monomial basis and tracks how many degree-raising nonlinear interactions are needed before discarded monomials can influence a chosen observable, such as a state coordinate. The resulting bounds are degree-aware, retain logarithmic-norm information from the original linearization, and yield geometric convergence over certified time horizons. This makes the truncation analysis sensitive to the actual degree path from the observable to the omitted tail, rather than to a worst-case global norm estimate (Boreale et al., 23 Jun 2026).

3. Basis changes, shifting, and globalization

The monomial basis is natural near an equilibrium when the vector field is polynomial-like, but it can be poorly matched to periodic or quasi-periodic structure. Carleman-Fourier linearization addresses this by replacing monomials with Fourier observables. For periodic vector fields,

$y_i:=x^{[i]},\qquad i\in\mathbb N,$2

the lifted variables are exponential modes rather than powers. In the multi-dimensional periodic setting of (Chen et al., 2024), the resulting operator is block upper triangular under an analyticity condition on the Fourier coefficients, and finite truncations approximate $y_i:=x^{[i]},\qquad i\in\mathbb N,$3 rather than $y_i:=x^{[i]},\qquad i\in\mathbb N,$4 directly. The 2025 quasi-periodic extension introduces an extended state $y_i:=x^{[i]},\qquad i\in\mathbb N,$5 and block variables built from exponentials $y_i:=x^{[i]},\qquad i\in\mathbb N,$6, recovering a block-upper-triangular hierarchy for multiple fundamental frequencies and proving exponential convergence of the primary block (Chen et al., 2024, Motee et al., 3 Mar 2025).

This basis change is not merely cosmetic. The Fourier lift removes the classical dependence on conditions such as $y_i:=x^{[i]},\qquad i\in\mathbb N,$7 and small initial data that arise in equilibrium-centered monomial theory. In the Kuramoto example of (Motee et al., 3 Mar 2025), the first block converges exponentially to $y_i:=x^{[i]},\qquad i\in\mathbb N,$8, and the approximation remains accurate over a broader range of initial phases and frequencies than the classical monomial truncation (Motee et al., 3 Mar 2025).

A different strategy is to change coordinates before lifting. The pivot-shifted framework of (Wang et al., 19 May 2026) replaces $y_i:=x^{[i]},\qquad i\in\mathbb N,$9 by $y_i'(t)=\sum_{j=0}^{k-1} A^i_{i+j} y_{i+j}(t).$0, then applies a Lyapunov transform $y_i'(t)=\sum_{j=0}^{k-1} A^i_{i+j} y_{i+j}(t).$1, and only then performs the Carleman lift. For quadratic ODEs

$y_i'(t)=\sum_{j=0}^{k-1} A^i_{i+j} y_{i+j}(t).$2

the shifted coefficients $y_i'(t)=\sum_{j=0}^{k-1} A^i_{i+j} y_{i+j}(t).$3 and $y_i'(t)=\sum_{j=0}^{k-1} A^i_{i+j} y_{i+j}(t).$4 can move the dynamics into a regime where the shifted linear part is stable. The paper proves long-time convergence when $y_i'(t)=\sum_{j=0}^{k-1} A^i_{i+j} y_{i+j}(t).$5 and derives short-time convergence guarantees even for unstable shifted systems, while removing the conventional lower-bound requirement on the initial condition (Wang et al., 19 May 2026).

Globalization methods attack a different weakness: the locality of a single expansion chart. The piecewise methods in (Novikau et al., 17 Oct 2025) reconstruct the Carleman embedding whenever the trajectory reaches the boundary of the current chart, either with fixed chart radius, adaptively changing radius, or with a static precomputed grid. The paper’s central claim is that standard Carleman embedding fails in regions where there are multiple fixed points, whereas piecewise charts make the method usable on systems ranging from multi-equilibrium one-dimensional flows to strange attractors (Novikau et al., 17 Oct 2025).

At the implementation level, (Akiba et al., 7 May 2026) studies duplicate-aware shift-and-lift assembly. By using symmetry-reduced monomial bases, packed exponent-key indexing, and sparse triplet coalescing, the lifted affine model

$y_i'(t)=\sum_{j=0}^{k-1} A^i_{i+j} y_{i+j}(t).$6

is assembled without proliferating duplicated monomial contributions. The same paper couples this assembly strategy to a moving-center expansion, so that shift and lift are updated jointly along the trajectory (Akiba et al., 7 May 2026).

4. Infinite-dimensional and PDE formulations

Carleman linearization extends beyond finite-dimensional ODEs. A direct PDE generalization is given in (Vaszary, 2024) for systems with quadratic nonlinearities. Instead of discrete monomial vectors, the lifted objects are tensor products of fields evaluated at independent copies of the spatial variable,

$y_i'(t)=\sum_{j=0}^{k-1} A^i_{i+j} y_{i+j}(t).$7

Differentiation yields a tridiagonal infinite linear PDE hierarchy in which each level couples only to levels $y_i'(t)=\sum_{j=0}^{k-1} A^i_{i+j} y_{i+j}(t).$8, $y_i'(t)=\sum_{j=0}^{k-1} A^i_{i+j} y_{i+j}(t).$9, and $\mathbf{x}=[x,x^2,x^3,\dots]^\mathsf T,$0. The paper emphasizes that truncation occurs in the Carleman hierarchy rather than by spatial discretization, and illustrates the construction on Burgers’ equation and a Vlasov-type system (Vaszary, 2024).

A more functional-analytic treatment is developed in (Heinzelreiter et al., 1 Oct 2025) for semilinear parabolic PDEs written as

$\mathbf{x}=[x,x^2,x^3,\dots]^\mathsf T,$1

The lifted moments $\mathbf{x}=[x,x^2,x^3,\dots]^\mathsf T,$2 satisfy a linear hierarchy with Kronecker-sum lifts of $\mathbf{x}=[x,x^2,x^3,\dots]^\mathsf T,$3, $\mathbf{x}=[x,x^2,x^3,\dots]^\mathsf T,$4, and $\mathbf{x}=[x,x^2,x^3,\dots]^\mathsf T,$5. The paper proves well-posedness of the truncated hierarchy in a Gelfand triple framework and, crucially, shows that after discretization the total approximation error decomposes into two independent components: the discretization error and the linearization error. This separation motivates structure-exploiting discretizations such as sparse grids and avoids mesh-dependent deterioration of the Carleman error estimate (Heinzelreiter et al., 1 Oct 2025).

Semigroup theory provides an even broader framework. In (Gakkhar et al., 5 May 2026), nonlinear evolution on a Hilbert space is embedded into a linear semigroup on

$\mathbf{x}=[x,x^2,x^3,\dots]^\mathsf T,$6

The Carleman operator is analyzed through dissipativity, Lumer–Phillips, and Trotter–Kato approximation. This replaces norm-ratio convergence conditions by semigroup generation and approximation conditions, and the paper treats unbounded operators through closability and integrated semigroup theory, with hyperviscous Burgers as a detailed example (Gakkhar et al., 5 May 2026).

The phrase “Carleman-based linearization” also appears in a related but non-identical PDE usage. The method of (Le et al., 2021) repeatedly linearizes an over-determined quasilinear elliptic PDE around the current iterate and solves the linearized subproblem via a Carleman-weighted quasi-reversibility functional. Its relation to classical Carleman linearization is conceptual rather than literal: the convergence mechanism comes from a Carleman estimate and successive linearization, not from monomial lifting (Le et al., 2021).

5. Applied uses of the lifted linear model

Carleman linearization is used less as an end in itself than as an enabling representation for analysis, control, approximation, and computation.

Area	Role of the lift	Representative paper
Nonlinear control	Bilinear approximation and Volterra kernels	(Bhatt et al., 2021)
Reachability	High-dimensional linear set propagation	(Forets et al., 2021)
System identification	Data-driven estimation of truncated lifted dynamics	(Abudia et al., 2022)
Chemical kinetics	Linearization for stiff reaction ODEs and HHL-compatible updates	(Akiba et al., 2022)
Polynomial recurrences	Infinite transition matrix for nonlinear recursions	(Myszkowski, 2021)
Fluid steady states	Second-order truncation as steady-state surrogate	(Cappelli et al., 22 May 2026)

In nonlinear control, (Bhatt et al., 2021) uses Carleman linearization to derive a bilinear model

$\mathbf{x}=[x,x^2,x^3,\dots]^\mathsf T,$7

then computes Volterra kernels up to third order and builds an IMC-Volterra controller. On the van de Vusse reactor benchmark, the third-order controller is reported to achieve the lowest control effort, with about a $\mathbf{x}=[x,x^2,x^3,\dots]^\mathsf T,$8 reduction in ISCI relative to the linear controller (Bhatt et al., 2021).

In reachability analysis, (Forets et al., 2021) lifts weakly nonlinear quadratic systems to a truncated linear ODE and propagates sets with support functions. Under weak nonlinearity

$\mathbf{x}=[x,x^2,x^3,\dots]^\mathsf T,$9

and dissipativity $c_m$0, the truncation error obeys

$c_m$1

which yields a dense-time overapproximation by bloating the projected lifted reach set with an error ball (Forets et al., 2021).

In system identification, (Abudia et al., 2022) uses measured trajectories to estimate the truncated lifted matrix directly by least squares,

$c_m$2

and selects the truncation order from a prescribed tolerance $c_m$3. For the Van der Pol example, the framework uses 209 trajectories sampled over $c_m$4 s and produces a certified finite-horizon trajectory bound of the form

$c_m$5

on $c_m$6 (Abudia et al., 2022).

In chemical kinetics, (Akiba et al., 2022) lifts polynomial reaction ODEs to a linear system $c_m$7, truncates at order $c_m$8, and discretizes implicitly: $c_m$9 The paper reports that increasing truncation order improves accuracy even with larger time steps, while the lifted dimension grows rapidly; in the H$f(0)=0$0/air case, increasing the truncation order by 1 increases the matrix size by about a factor of 9 (Akiba et al., 2022).

Carleman linearization also applies to nonlinear recurrences. In (Myszkowski, 2021), polynomial recurrences are embedded into infinite transition matrices whose powers yield explicit solution formulas once the matrix is shifted to an upper-triangular and diagonalizable form. The method covers uni-variable depth-one recurrences, multi-variable depth-one systems, and arbitrary finite-depth systems after auxiliary-variable reduction (Myszkowski, 2021).

For dissipative fluid problems, (Cappelli et al., 22 May 2026) studies the lowest nontrivial truncation $f(0)=0$1, denoted C2. The paper’s main claim is that C2 can recover the late-time steady state even when it misses transient detail. This is shown analytically for a forced logistic equation and numerically for moderate-Re two-dimensional flows, suggesting that very low-order Carleman truncations may still preserve asymptotic structure (Cappelli et al., 22 May 2026).

6. Quantum formulations, computational limits, and current outlook

A major contemporary motivation for Carleman linearization is that quantum algorithms are much better developed for linear systems than for nonlinear ones. In chemical kinetics, (Akiba et al., 2022) uses the HHL motivation directly: nonlinear reaction ODEs are first linearized, then rewritten as linear systems of the form

$f(0)=0$2

In the broader quantum-simulation setting of (Wu et al., 2024), the truncation level needed for error $f(0)=0$3 scales as

$f(0)=0$4

under a non-resonance condition, which is the key route to quantum advantage beyond strictly dissipative dynamics (Wu et al., 2024).

Quantum implementations also expose structural constraints of the lift itself. The local Carleman linearization algorithm for quantum lattice Boltzmann dynamics in (Zamora et al., 17 Nov 2025) uses second-order lifting with variables $f(0)=0$5 and $f(0)=0$6, reorganizes the encoding so that the collision remains local in the quantum registers, and reports per-time-step cost

$f(0)=0$7

with a correct-branch probability of order $f(0)=0$8. This is a concrete illustration that truncation order, encoding locality, and nonunitary embedding overhead are inseparable in quantum Carleman algorithms (Zamora et al., 17 Nov 2025).

The semigroup and pivot-shifted lines of work suggest two distinct strategies for enlarging the class of tractable systems. The semigroup approach of (Gakkhar et al., 5 May 2026) replaces fragile norm-ratio arguments by dissipativity and Trotter–Kato approximation, while the pivot-shifted framework of (Wang et al., 19 May 2026) stabilizes the coordinates before lifting and proves logarithmic truncation-order dependence on simulation time and precision in the stable shifted regime. The latter is designed specifically for preparing a quantum state proportional to the final solution of a quadratic ODE (Gakkhar et al., 5 May 2026, Wang et al., 19 May 2026).

The limitations remain explicit in the recent literature. Truncation error is unavoidable; higher-order truncations improve accuracy but rapidly increase dimension; local convergence can fail near multiple fixed points; closure and conditioning can dominate performance in high-order or moving-center schemes; and low-order models may preserve steady states while missing transient detail. This suggests that the current frontier is not a single universal Carleman theory, but a family of embeddings whose usefulness depends on how well the lifted basis, coordinate choice, and truncation scheme match the structure of the nonlinear system under study (Novikau et al., 17 Oct 2025, Akiba et al., 7 May 2026, Cappelli et al., 22 May 2026).