Finite-data error bounds for Koopman-based prediction and control

1. Introduction

• Koopman methods predict nonlinear-system evolution by propagating observables with a linear, infinite-dimensional operator rather than simulating the nonlinear system directly.The introduction links their recent success to numerically tractable approximation techniques such as eDMD.Source: §1 · p. 2
• The introduction identifies [17] as the only prior rigorous finite-data error-bound analyses for Koopman approximations of nonlinear ODEs.Other cited work left finite-data approximation error unquantified or did not derive prediction-error bounds.Source: §1 · p. 2
• The paper derives probabilistic finite-data bounds for Koopman approximation and multi-step prediction errors for nonlinear SDEs, including ODEs, with bounds explicitly depending on data size.The analysis uses mass concentration inequalities and numerical error analysis for error propagation, and adds an ergodic-sampling assumption tailored to stationary SDEs.Source: §1 · p. 2
• For nonlinear control-affine systems, the analysis bounds approximation errors for the projected generator, semigroup, and trajectories, as well as open-loop prediction accuracy.The approach uses bilinear surrogate models built from nc + 1 autonomous Koopman operators without augmenting the state by control inputs, and its prediction estimates can apply to MPC.Source: §1 · p. 3

2. Finite-data bounds on the approximation error: nonlinear SDEs

• The analysis covers finite-dimensional SDEs with Lipschitz drift and diffusion, while setting the diffusion to zero recovers the ordinary differential equation case.The assumptions include either a compact forward-invariant state space with normalized Lebesgue measure or an invariant probability measure.Source: §2 · pp. 3–4

2.1. Extended Dynamic Mode Decomposition

• eDMD approximates the Koopman generator by Galerkin-projecting it onto a finite-dimensional observable space, using empirical mass and stiffness matrices from ergodic-trajectory or i.i.d. data.Under the stated sampling assumptions, the empirical estimator is well-defined for sufficiently large samples with probability one.Source: §2.1 · pp. 5–6

2.2.1. Concentration bounds for random matrices

• Ergodic-sampling concentration bounds separate an asymptotic variance contribution from a data-dependent contribution that decays polynomially with sample size.These scalar variance estimates are extended to matrix-valued quantities, yielding probabilistic concentration bounds; the i.i.d. case has simpler bounds.Source: §2.2.1 · pp. 7–9

2.2.2. Error bound for the projected generator

• Theorem 12 bounds the projected-generator approximation error probabilistically for ergodic, reversible ergodic, and i.i.d. sampling, with sample requirements depending on the error and tolerance.The bounds apply under Assumption 5 and distinguish the three sampling settings.Source: §2.2.2 · pp. 9–10
• For any finite prediction horizon and error and probability tolerances, sufficiently many data points bound the prediction error between the projected and data-driven generator trajectories.The estimate can be refined or made uniform in time under additional assumptions such as stability of the Koopman semigroup generated by the projected operator.Source: §2.2.2 · p. 11

2.3. Error bound for the projected Koopman operator

• For sampling times that are integer multiples of the snapshot interval, the section sketches an analogous probabilistic error-bound approach for the projected Koopman operator.The supplied passage introduces the argument but does not state its resulting bound.Source: §2.3 · p. 12
• Theorem 14 gives a finite-data result under Assumption 5 for any t ≥ 0, error bound ε > 0, and probabilistic tolerance δ ∈ (0, 1).The supplied theorem passage ends before stating the resulting bound.Source: §2.3 · p. 12
• The sufficient sample threshold m0 can be specified analogously to the threshold in Theorem 12.Source: §2.3 · p. 12

3. Extension to control systems

• Control affinity of the Koopman generator lets the authors combine drift and input-direction generators to form a bilinear control system in the observables.The analysis first treats autonomous systems driven by constant controls, then uses control affinity to extend the estimates to general controls.Source: §3 · pp. 13–14
• The data assumptions allow either i.i.d. samples or data from a single ergodic trajectory, with the latter requiring exponential stability of the Koopman semigroup.The i.i.d. case includes sampling from normalized Lebesgue measure on a compact set or from an invariant measure.Source: §3 · p. 13
• For each constant control, Proposition 16 gives a probabilistic approximation-error estimate once the sample count exceeds a threshold determined by the target error and tolerance.The minimum sample count is specified using Theorem 12.Source: §3 · p. 14
• Theorem 17 extends the generator estimates to controls in L∞(0, T; R^nc) when the data count exceeds the largest threshold across the constant-control cases.The result applies for a chosen prediction horizon and probabilistic tolerance.Source: §3 · p. 14
• The trajectory bound follows via Gronwall when the state response remains in X, and Corollary 18 assumes this containment with probability one.Source: §3 · p. 15
• For a locally Lipschitz cost and an optimal response contained in X, sufficiently many data points yield a feasible surrogate tuple with an estimated cost difference.The corollary compares the projected Koopman optimal-control problem with the surrogate-based control problem.Source: §3 · p. 16

4.1. Generator Error Bounds: Ornstein-Uhlenbeck Process

• For the OU process, entry-wise generator-matrix bounds overestimate errors by only 2–3×, while Frobenius-norm bounds differ by about an order of magnitude.The qualitative behavior of all theoretical bounds is confirmed by the data; actual Frobenius error only marginally exceeds the maximal entry-wise error.Source: §4.1 · pp. 17–18, Figure 1 · p. 17

4.2. Extension to control systems

• The control examples compare exact-model solutions with bilinear surrogates and eDMDc, with project-and-lift used where needed for prediction.The authors do not address theoretical optimality of computed trajectories because the available error bounds are still too large.Source: §4.2 · p. 18

4.2.1. Duffing equation (ODE)

• For Duffing prediction, the bilinear surrogate’s relative error stays below 0.1% for almost 3 seconds, while eDMDc starts near 10% error and becomes unstable within one second.Figure 2 compares the ODE solution, bilinear surrogate, and eDMDc model under a random control input.Source: §4.2.1 · pp. 19–20, Figure 2 · p. 19
• Duffing one-step training error decreases with more data but saturates by m = 30; the study reports no further improvement beyond that size.The boxplots summarize 20 training runs.Source: §4.2.1 · p. 20, Figure 3 · p. 20
• In the Duffing control case, smaller lag times improve accuracy because the bilinear surrogate is exact only for the generator and is first-order for finite lag.Accuracy still significantly exceeds that of eDMDc.Source: §4.2.1 · p. 20
• Duffing prediction performs well with monomials of degree three or higher; adding roots does not help, and smaller dictionaries reduce data requirements.The authors report that this agrees with their error analysis and prior work.Source: §4.2.1 · p. 20, Figure 4 · p. 21

4. Numerical examples

• For Duffing control, the bilinear surrogate nearly matches the true model, while eDMDc fails; for controlled OU, the bilinear model performs well with m = 100 training points.The OU prediction study also reports comparable performance when control enters linearly, but very poor eDMDc performance otherwise.Source: §4.2.1 · p. 21, Figure 5 · p. 21, §4.2.2 · p. 22

4.2.2. Ornstein-Uhlenbeck process (SDE)

• For controlled OU prediction, expected values are estimated by averaging 100 simulations, and the study reports no considerable accuracy deterioration even in the very low-data limit.In control, the bilinear surrogate performs well with m = 100 training points, while 20 simulations per objective evaluation proved insufficient for adequate full-system performance.Source: §4.2.2 · p. 22, Figure 6 · p. 22, Figure 7 · p. 23

5. Conclusions

• The paper presents the first rigorously derived probabilistic finite-data approximation bounds for Koopman generators of SDEs and nonlinear control systems.The bounds explicitly depend on the number of data points.Source: §5 · p. 23
• The analysis estimates prediction accuracy in terms of data size by accounting for error propagation.Source: §5 · p. 23
• For ODEs, the paper relaxes the i.i.d.-data assumption used in using more advanced probability techniques.Source: §5 · p. 23
• The techniques provide a theoretical foundation for a Koopman-based control-affine approach that avoids augmenting the state by control inputs and is described as well-suited for MPC.The paper says this avoids the curse of dimensionality with respect to control dimension.Source: §5 · p. 23

A.1. Norm of the isomorphism V ≃Rn

• The subsection considers B as an operator in L(V, V) and its corresponding matrix representation in R^n×n.Source: §A.1 · p. 26
• An identity is used to show equivalence of the vector norms.Source: §A.1 · p. 26
• The vector-norm equivalence induces equivalence of the corresponding operator norms.Source: §A.1 · p. 26

A.2. A technical lemma

• Lemma 21 concerns a collection of measurable sets A_i.Source: §A.2 · p. 26
• When each event has probability at least 1−δ, the proof bounds the first sum below by d(1−δ) and the second by −(d−1).Source: §A.2 · p. 26
• The proof applies a classical formula inductively to establish the lemma's first claim.Source: §A.2 · p. 26

A.3. Proof of the error bound on the trajectories

• Lemma 22 considers solutions z and z̃ of equations (9) and (10), respectively.Source: §A.3 · p. 27
• The proof forms the error e = z − z̃ by subtracting equation (10) from equation (9).Source: §A.3 · p. 27
• Integrating the difference over [0, t] yields an intermediate trajectory-error relation.Source: §A.3 · p. 27
• Gronwall's inequality is used to obtain the trajectory bound, which is then applied in the proof of Corollary 13.Source: §A.3 · p. 27

• The claim is obtained by applying Theorem 12 together with the inequality ∥·∥2 ≤∥·∥F.Source: We compute · p. 27
• A threshold m0 depends on T, z0, LV, and ε, and the probability statement holds for any t ≥0.Source: We compute · p. 27
• Taking the minimum over t ∈[0, T] completes the claim.Source: We compute · p. 27
• Corollary 23 states a result under the condition that the Koopman semigroup generated by LV is bounded by M.Source: We compute · p. 28

• If the semigroup generated by ˜Lm is exponentially stable, ∥˜z(t) −z(t)∥2 can be bounded uniformly for t ≥0.Source: If it is exponentially stable then · p. 28
• For 1 ≤p ≤∞, the stated bound uses M ≥1 and c = c(p) ≥0 independent of t.Source: If it is exponentially stable then · p. 28
• The proof defines the error as e(t) = ˜z(t) −z(t) by subtracting equations (10) and (9).Source: If it is exponentially stable then · p. 28
• Variation of constants with the Koopman semigroup generated by LV is used to derive the error estimate.Source: If it is exponentially stable then · p. 28
• Exponential stability gives ∥˜z(t)∥2 ≤˜ Me−˜ωt∥z0∥2, which makes the upper bound uniform in time.Source: If it is exponentially stable then · p. 28

• The generator eigenvalues are κl = −l, the Koopman eigenvalues are λl(t) = e−lt, and the eigenfunctions are scaled physicist’s Hermite polynomials orthonormal under a normal density of variance one half.Source: §A.4 · pp. 28–29
• The monomial basis is recovered from the eigenfunction basis through a representation formula.Source: §A.4 · p. 29
• For monomials up to degree N, Galerkin matrices C and A are obtained from moments of a normal distribution with variance 0.5, with odd-degree moments zero.Source: §A.4 · p. 29
• The asymptotic variance calculation centers φij and uses n = i + j for C or n = i + j −2 for A; centering removes the H0 contribution.Source: §A.4 · p. 29
• Lag-time matrix elements are calculated using orthogonality, then Lemma 6 gives the asymptotic variance with qk = e−(n−2k)∆t and the l = 0 contribution counted once.Source: §A.4 · p. 29