carleman-linearization-ode-solver

name: carleman-linearization-ode-solver description: "Carleman linearization methodology for converting nonlinear ODEs into infinite-dimensional linear systems via tensor powers. C2 (2nd order truncation) recovers both transient and steady-state solutions. Use when: (1) solving nonlinear differential equations, (2) quantum algorithms for ODEs (HHL-based), (3) fluid dynamics steady-state approximation, (4) converting nonlinear systems to linear form for quantum computation, (5) numerical analysis of dynamical systems. Keywords: carleman linearization, ODE solver, nonlinear differential equations, quantum ODE, fluid flow simulation, C2 truncation" metadata: arxiv_id: "2605.23380" published: "2026-05-26" tags: [numerical-analysis, differential-equations, quantum-algorithms, fluid-dynamics, carleman-linearization]

Carleman Linearization for ODE Solving

Core Concept

Carleman linearization converts a system of nonlinear ODEs into an infinite-dimensional linear system by embedding the state into tensor powers. The key insight: nonlinear terms like x², x³ become linear in higher-dimensional space.

C2 (2nd order truncation): Truncating at 2nd order recovers both the initial transient AND the steady-state solution — a surprising asymptotic property proved analytically for decaying logistic equations and verified for 2D fluid flows at moderate Reynolds numbers.

Mathematical Framework

For a nonlinear ODE system:

dx/dt = A₁x + A₂(x⊗x) + A₃(x⊗x⊗x) + ...

Carleman linearization lifts to infinite dimensions:

dX/dt = M·X

where X = [x, x⊗x, x⊗x⊗x, ...]ᵀ and M is an infinite block-upper-triangular matrix. The 2nd-order truncation (C2) keeps only x and x⊗x levels.

Key Findings

1. C2 captures steady-state: The 2nd-order truncation recovers not just transient dynamics but also the late-time steady-state solution
2. Analytical proof: Proven for decaying logistic equation with external forcing
3. Empirical validation: Holds for 2D fluid flows at moderate Reynolds numbers
4. Quantum relevance: Carleman linearization is the primary method for encoding nonlinear ODEs into quantum algorithms (HHL-based solvers)

Usage Patterns

Pattern 1: Nonlinear ODE → Linear System

Convert a nonlinear system for quantum or classical linear solver:

1. Identify the nonlinear ODE: dx/dt = f(x)
2. Decompose f(x) into polynomial terms
3. Construct the Carleman embedding matrix M
4. Truncate at desired order (C2 for steady-state, higher for accuracy)
5. Solve the linear system dX/dt = M·X

Pattern 2: Quantum Algorithm for Nonlinear ODEs

For quantum computation of nonlinear dynamics:

1. Apply Carleman linearization to convert nonlinear ODE to linear system
2. Discretize in time (finite difference)
3. Encode as linear system Ax = b
4. Apply HHL algorithm or variants
5. Extract solution from quantum state

Pattern 3: Fluid Flow Steady-State Approximation

For fluid dynamics at moderate Reynolds numbers:

1. Write Navier-Stokes in discretized form
2. Apply C2 linearization (2nd order truncation)
3. Solve the resulting linear system for steady-state
4. Verify convergence against full nonlinear simulation

When to Use C2 vs Higher Orders

Error Handling

Dimensionality Explosion

Higher-order Carleman truncations cause exponential growth in system size. For n variables at order k: dimension = O(n^k). Mitigation: Use C2 as baseline; only increase order if residual error is unacceptable.

Convergence Domain

Carleman linearization requires the solution to remain within the convergence radius of the Taylor series. Mitigation: Monitor solution norm; apply rescaling if approaching divergence boundary.

Quantum Algorithm Limitations

HHL-based quantum ODE solvers require well-conditioned matrices. Carleman-embedded matrices may be ill-conditioned. Mitigation: Apply preconditioning; consider alternative quantum ODE algorithms (e.g., SLAC derivatives).

Related Skills

• carleman-vqls — Carleman linearization + VQLS for quantum linear solvers
• dolq-ode-discovery-llm — ODE discovery with LLMs
• pem-ude-neural-governing-equations — Governing equation discovery
• ode-complexity-dynamics — ODE complexity analysis