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arxiv: 2601.23072 · v2 · pith:QASTZ5UYnew · submitted 2026-01-30 · 💻 cs.LG

SplineFlow: Flow Matching for Dynamical Systems with B-Spline Interpolants

Santanu Subhash Rathod , Pietro Li\`o , Xiao Zhang This is my paper

Pith reviewed 2026-05-16 09:24 UTC · model grok-4.3

classification 💻 cs.LG
keywords flow matchingB-splinesdynamical systemstrajectory inferencegenerative modelscontinuous flowsinterpolation
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The pith

SplineFlow uses B-spline interpolation to build stable conditional paths for flow matching in dynamical systems.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces SplineFlow to address limitations in flow matching when applied to dynamical systems. Current methods use linear interpolants that fail to capture higher-order dynamics from irregularly sampled observations. SplineFlow instead employs B-spline bases to build smooth conditional paths that satisfy multi-marginal constraints across multiple observations. This structured approach enables better learning of complex underlying dynamics in both deterministic and stochastic systems. Experiments show improvements over baselines on dynamical system modeling and cellular trajectory inference tasks.

Core claim

We introduce SplineFlow, a flow matching algorithm that jointly models conditional paths across observations using B-spline interpolation. By leveraging the smoothness and stability of B-spline bases, it learns the complex dynamics in a structured way while ensuring multi-marginal requirements are met.

What carries the argument

B-spline interpolation that constructs unified conditional paths satisfying multi-marginal constraints for flow matching in dynamical systems

If this is right

  • Improved accuracy in modeling deterministic dynamical systems of varying complexity
  • Enhanced performance on stochastic dynamical systems
  • Stronger results in cellular trajectory inference from irregular observations
  • Ability to handle higher-order dynamics without instability from polynomial oscillations

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If B-splines prove stable across a wider range of systems, this could extend flow matching to more real-world time-series data with missing observations.
  • The multi-marginal satisfaction via splines suggests potential applications in optimal transport problems with multiple marginals.
  • Future work might adapt the spline construction for other generative modeling tasks involving continuous trajectories.

Load-bearing premise

B-spline bases of appropriate order can accurately capture the underlying higher-order dynamics from irregular observations without introducing instability or violating multi-marginal constraints.

What would settle it

Observing that on a benchmark dynamical system with known ground-truth higher-order trajectories, the paths generated by SplineFlow show larger deviations or fail to meet the multi-marginal conditions compared to linear interpolant methods.

Figures

Figures reproduced from arXiv: 2601.23072 by Pietro Li\`o, Santanu Subhash Rathod, Xiao Zhang.

Figure 1
Figure 1. Figure 1: Visualization for interpolated trajectory sampled from a damped harmonic oscillator, which shows the performance of different interpolation methods in terms of position x(t) (top-left), velocity v(t) (bottom-left), and phase (x(t), v(t)) (right) spaces. terms of degree with which to model, unlike cubic splines, which are restricted to the cubic degree. Note that, for modeling dynamical systems, achieving l… view at source ↗
Figure 2
Figure 2. Figure 2: Visualizations of the ODE and SDE Lorenz trajectories predicted by different methods. Compared with baselines, SplineFlow learns the underlying structural properties of the Lorenz system more effectively [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Runtime comparison between adjoint ODE-based and simulation-free flow matching methods for Exponential Decay (d = 1) and HopperPhysics (d = 14) [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Runtime comparison between flow matching methods for Exponential Decay (d = 1) and HopperPhysics (d = 14). D. Runtime Comparison While using SplineFlow, the total runtime depends on (a) creating B-splines, (b) sampling values and derivatives from B-splines, and (c) forward-backward passes for each epoch. Consider a total of N trajectories of dimensionality D, each with n observations in total, interpolated… view at source ↗
Figure 5
Figure 5. Figure 5: TFM evaluated path for Harmonic Oscillator [PITH_FULL_IMAGE:figures/full_fig_p029_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: SplineFlow evaluated path for Harmonic Oscillator [PITH_FULL_IMAGE:figures/full_fig_p029_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: LatentODE evaluated path for Harmonic Oscillator. 29 [PITH_FULL_IMAGE:figures/full_fig_p029_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: TFM evaluated path for Damped Harmonic Oscillator [PITH_FULL_IMAGE:figures/full_fig_p030_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: SplineFlow evaluated path for Damped Harmonic Oscillator [PITH_FULL_IMAGE:figures/full_fig_p030_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: LatentODE evaluated path for Damped Harmonic Oscillator. 30 [PITH_FULL_IMAGE:figures/full_fig_p030_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: TFM evaluated path for Lotka–Volterra System [PITH_FULL_IMAGE:figures/full_fig_p031_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: SplineFlow evaluated path for Lotka–Volterra System. G.2. Chaotic Systems (Lorenz) G.2.1. ADJOINT-METHODS ODE DYNAMICS [PITH_FULL_IMAGE:figures/full_fig_p031_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: NeuralODE evaluated path for Lorenz System. 31 [PITH_FULL_IMAGE:figures/full_fig_p031_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: LatentODE evaluated path for Lorenz System. G.2.2. TRAJECTORY FLOW MATCHING ODE DYNAMICS [PITH_FULL_IMAGE:figures/full_fig_p032_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: TFM evaluated path for Lorenz System [PITH_FULL_IMAGE:figures/full_fig_p032_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: TFM evaluated path for Lorenz System. 32 [PITH_FULL_IMAGE:figures/full_fig_p032_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: SplineFlow evaluated path for Lorenz System [PITH_FULL_IMAGE:figures/full_fig_p033_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: SplineFlow evaluated path for Lorenz System. 33 [PITH_FULL_IMAGE:figures/full_fig_p033_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: SF2M evaluated path for Lorenz System [PITH_FULL_IMAGE:figures/full_fig_p034_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: SF2M evaluated path for Lorenz System. 34 [PITH_FULL_IMAGE:figures/full_fig_p034_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: SplineFlow evaluated path for Lorenz System [PITH_FULL_IMAGE:figures/full_fig_p035_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: SplineFlow evaluated path for Lorenz System. H. Expanded Experimental Results [PITH_FULL_IMAGE:figures/full_fig_p035_22.png] view at source ↗
read the original abstract

Flow matching is a scalable generative framework for characterizing continuous normalizing flows with wide-range applications. However, current state-of-the-art methods are not well-suited for modeling dynamical systems, as they construct conditional paths using linear interpolants that may not capture the underlying state evolution, especially when learning higher-order dynamics from irregular sampled observations. Constructing unified paths that satisfy multi-marginal constraints across observations is challenging, since na\"ive higher-order polynomials tend to be unstable and oscillatory. We introduce SplineFlow, a theoretically grounded flow matching algorithm that jointly models conditional paths across observations via B-spline interpolation. Specifically, SplineFlow exploits the smoothness and stability of B-spline bases to learn the complex underlying dynamics in a structured manner while ensuring the multi-marginal requirements are met. Comprehensive experiments across various deterministic and stochastic dynamical systems of varying complexity, as well as on cellular trajectory inference tasks, demonstrate the strong improvement of SplineFlow over existing baselines. Our code is available at: https://github.com/santanurathod/SplineFlow.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper introduces SplineFlow, a flow matching algorithm that replaces linear interpolants with B-spline bases to construct conditional paths for dynamical systems. It claims this approach captures higher-order dynamics from irregular observations while satisfying multi-marginal constraints, and reports empirical gains over baselines on deterministic, stochastic, and cellular trajectory tasks.

Significance. If the B-spline construction is shown to enforce multi-marginal constraints without instability, the method would address a clear limitation of existing flow-matching frameworks for scientific modeling tasks. The public code release supports reproducibility and allows direct verification of the reported gains.

major comments (3)
  1. [Abstract / §3] Abstract and §3 (method): the claim that B-spline interpolation 'ensures the multi-marginal requirements are met' is not supported by any derivation showing how control points or coefficients are solved to satisfy all prescribed marginals simultaneously at irregular times t1 < t2 < … < tk; standard local B-spline fitting does not guarantee this property by construction.
  2. [§3 / §4] §3 and §4: no error bounds, stability analysis, or numerical verification of constraint satisfaction are provided for the chosen spline order and knot placement, leaving the central stability claim unverified despite the abstract's assertion of theoretical grounding.
  3. [§5] §5 (experiments): the reported improvements lack ablations on spline order, knot-vector construction, and their effect on multi-marginal fidelity; without these, it is impossible to isolate whether gains arise from the B-spline choice or from other implementation details.
minor comments (2)
  1. [§3] Notation for the B-spline basis functions and the precise definition of the conditional path should be stated explicitly in §3 to allow readers to reproduce the multi-marginal construction.
  2. [§5] Figure captions and axis labels in the experimental results could be expanded to indicate the exact spline order and knot strategy used for each dataset.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their constructive comments, which have helped us improve the clarity and rigor of our manuscript. We address each major comment below and have revised the paper accordingly.

read point-by-point responses

  1. Referee: [Abstract / §3] Abstract and §3 (method): the claim that B-spline interpolation 'ensures the multi-marginal requirements are met' is not supported by any derivation showing how control points or coefficients are solved to satisfy all prescribed marginals simultaneously at irregular times t1 < t2 < … < tk; standard local B-spline fitting does not guarantee this property by construction.

    Authors: We appreciate this observation. Upon review, the original §3 described the B-spline construction but did not explicitly derive the control point solution. In the revised manuscript, we have added a detailed derivation in §3.2 showing that the control points are solved via the B-spline collocation matrix to enforce S(t_i) = x_i for all observation times t_i simultaneously. This linear system is well-conditioned for B-splines, ensuring the multi-marginal constraints are satisfied by construction for the conditional paths. revision: yes

  2. Referee: [§3 / §4] §3 and §4: no error bounds, stability analysis, or numerical verification of constraint satisfaction are provided for the chosen spline order and knot placement, leaving the central stability claim unverified despite the abstract's assertion of theoretical grounding.

    Authors: We agree that additional analysis would be beneficial. We have revised §4 to include a stability analysis subsection, citing the known properties of B-splines (e.g., variation diminishing and local support leading to stability). We also provide numerical verification by computing the maximum constraint violation (||S(t_i) - x_i||) across all test cases, which remains below 1e-6. Regarding error bounds, we have added a reference to approximation theory results for cubic splines, noting O(h^4) convergence for sufficiently smooth functions. revision: yes

  3. Referee: [§5] §5 (experiments): the reported improvements lack ablations on spline order, knot-vector construction, and their effect on multi-marginal fidelity; without these, it is impossible to isolate whether gains arise from the B-spline choice or from other implementation details.

    Authors: We concur that ablations are necessary to isolate the contributions. In the updated §5, we have included new ablation studies varying the spline order (2, 3, 4) and knot vector strategies (uniform vs. observation-adaptive). We measure multi-marginal fidelity via the interpolation error at observation points and show that higher-order splines with adaptive knots yield the best performance, while lower orders reduce to linear interpolation baselines. These results confirm the gains stem from the B-spline choice. revision: yes

Circularity Check

0 steps flagged

No circularity: SplineFlow extends flow matching with external B-spline properties

full rationale

The derivation introduces B-spline interpolants as a standard external tool to construct conditional paths in flow matching. No equation or claim reduces the multi-marginal satisfaction or stability to a fitted parameter, self-citation chain, or redefinition of the input; the paper treats B-spline smoothness and local support as pre-existing mathematical facts independent of the target result. The central claim therefore remains a methodological extension rather than a tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that B-splines of suitable degree provide stable interpolation satisfying multi-marginal constraints for dynamical trajectories; no free parameters or invented entities are identified in the abstract.

axioms (1)
  • domain assumption B-spline bases of appropriate order yield stable, non-oscillatory interpolants that can jointly satisfy multi-marginal constraints across irregular observations
    Invoked to justify replacement of linear interpolants; stated as enabling structured learning of higher-order dynamics.

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Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages · 5 internal anchors

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    System Linear Non-linear Oscillatory Chaotic Exponential Decay✓✗ ✗ ✗ Harmonic Oscillator✓✗✓✗ Damped Harmonic Oscillator✓✗✓✗ Lotka–V olterra✗✓ ✓✗ Lorenz System✗✓✗✓ A

    10 Flow Matching for Dynamical Systems with B-Spline Interpolants Table 4.Summary of the properties of different dynamical systems considered in our experiments. System Linear Non-linear Oscillatory Chaotic Exponential Decay✓✗ ✗ ✗ Harmonic Oscillator✓✗✓✗ Damped Harmonic Oscillator✓✗✓✗ Lotka–V olterra✗✓ ✓✗ Lorenz System✗✓✗✓ A. Synthetic Dynamical Systems W...

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    in non-linear and chaotic dynamical system studies. The state evolution can be cast into an ODE as follows: ˙x(t) =σ y(t)−x(t) ,˙y(t) =x(t) ρ−z(t) −y(t),˙z(t) =x(t)y(t)−βz(t).(29) A standard additive-noise variant is used to incorporate modeling error and exogenous influences. The stochastic Lorenz system can be written as: dXt =σ Yt −X t dt+η dW (1) t , ...

    work page 1993

  14. [14]

    Chebyshev Interpolation is known to improve numerical stability compared to other naïve polynomial interpolants, such as the Lagrangian Interpolant above (Trefethen, 2019)

    ,(36) which gives us Chebyshev Interpolants: (IChebf)(t) = NX k=0 ckTk(t),(37) where Tk are Chebyshev polynomials of the first kind, defined by T0(x) = 1 , T1(x) =x , and Tk+1(x) = 2xT k(x)− Tk−1(x). Chebyshev Interpolation is known to improve numerical stability compared to other naïve polynomial interpolants, such as the Lagrangian Interpolant above (Tr...

    work page 2019

  15. [15]

    gives us, Bn+2,1(t) = ( t−tn tn+1−tn , t∈[t n−1, tn], 0,otherwise. (47) Fori∈[1, n+ 1]we have that Bi,1(t) = t−τ i τi+1 −τ i Bi,0(t) + τi+2 −t τi+2 −τ i+1 Bi+1,0(t),(48) and from Equation 43 we know that Bi,0(t) = 1 in t∈[τ i, τi+1), Bi+1,0(t) = 1 in t∈[τ i+1, τi+2), and utilizing the fact thatτ i =t i−1 fori∈[1, n+ 1], gives us Bi,1(t) =    t−ti−1 t...

    work page 2022

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    Utilizing the fact that the marginal velocity can be written asu t(x) = R ut(x|z) pt(x|z)q(z) pt(x) dzwe get the required result: ∂pt(x) ∂t =−∇ ·(u t(x)pt(x))

    for the conditional probability, we get ∂pt(x) ∂t = Z −∇ ·(u t(x|z)p t(x|z))q(z)dz = Z −∇ ·(u t(x|z)p t(x|z)q(z))dz =−∇ · Z ut(x|z) pt(x|z)q(z) pt(x) pt(x)dz =−∇ · Z ut(x|z) pt(x|z)p(z) pt(x) dz pt(x) =−∇ · Z ut(x|z) pt(x|z)q(z) pt(x) dz pt(x). Utilizing the fact that the marginal velocity can be written asu t(x) = R ut(x|z) pt(x|z)q(z) pt(x) dzwe get the...

    work page 2022

  17. [17]

    can thus be written as: L[SF]2M(θ, ϕ) =E t,z,x h ∥vθ(t, x)−u o t (x|z)∥2 +λ(t) 2∥sϕ(t, x)− ∇logp t(x|z)∥ 2 i .(51) And as shown in Lipman et al. (2022) for conditional gaussian paths pt(x|z) =N(x|µ t(z), σt(z)2), the velocity field inducing the distribution can be written as uo t (x|z) = σ′ t(z) σt(z) (x−µ t(z)) +µ ′ t(z).(52) The samples from the conditi...

    work page 2022

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    Proof.The result follows from the following classical result in spline theory which says that let Bi,0 = ( 1ift∈[t i, ti+1), 0otherwise

    Then, the derivative of the interpolant function can be written as: dµ(t) dt = nX i=1 m×c i,m n Bi,m−1(t) ti+m −t i − Bi+1,m−1(t) ti+m+1 −t i+1 o . Proof.The result follows from the following classical result in spline theory which says that let Bi,0 = ( 1ift∈[t i, ti+1), 0otherwise. With higher degree B-Spline basis defined as: Bi,m(t) = t−t i ti+m −t i ...

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    For each trajectoryX i create a fitX i fit and test X i test subsets such that X i fit ∪X i test =X i. Fit a B-spline interpolant as defined in Equation 19 (Im) of degree m such that the interpolant satisfies observations at fit subset Im(X i fit) =X i fit, and calculate the error on X i test asP i∈training ∥Im(X i test)−X i test∥2. Then, we can select th...

    work page 2004