Diffeomorphic B-spline Flows

• Diffeomorphic B-spline flows are smooth, invertible transformations parameterized by B-spline functions that ensure continuity and stability.
• They leverage the local support and analytic inversion of B-spline bases to model complex static distributions and dynamic systems efficiently.
• Rigorous derivative constraints guarantee diffeomorphism and bi-Lipschitz continuity, offering state-of-the-art interpolation and sampling in practical applications.

A diffeomorphic B-spline flow is a type of normalizing flow where the invertible transformation is parameterized via (possibly non-uniform) B-spline functions and is constructed to be diffeomorphic, i.e., smooth and invertible with a smooth inverse. These models leverage the local support and high smoothness of B-spline bases to define continuous, non-oscillatory, and stable flows, with rigorous conditions ensuring diffeomorphism and (bi-)Lipschitz continuity. Diffeomorphic B-spline flows have been developed both for modeling static probability distributions and for learning complex dynamical systems from sparse or irregular observations, as exemplified by the SplineFlow and Neural Diffeomorphic Non-uniform B-spline Flow frameworks (Rathod et al., 30 Jan 2026, Hong et al., 2023).

1. Mathematical Foundations

Diffeomorphic B-spline flows generalize the notion of continuous normalizing flows (CNFs) and discrete coupling-based flows by introducing transformations parameterized by B-splines. In the CNF setting, the state $z_t \in \mathbb{R}^d$ evolves according to an ODE:

$\frac{d z_t}{dt} = v_t(z_t), \quad t \in [0,1]$

with $v_t$ a time-dependent vector field parameterized, for example, by a neural network. If $v_t$ is globally Lipschitz, by classical ODE theory the time-$t$ map $z_0 \mapsto z_t$ is a $\mathcal{C}^1$ diffeomorphism (Rathod et al., 30 Jan 2026). For coupling-based flows, the transformation in each (possibly one-dimensional) channel is parameterized as a B-spline function, leading to a piecewise polynomial and highly smooth (at least $\mathcal{C}^{k-2}$ for B-splines of order $k$) invertible transformation (Hong et al., 2023).

Let $\{t_j\}$ be (possibly non-uniform, possibly periodic) knots and $\frac{d z_t}{dt} = v_t(z_t), \quad t \in [0,1]$0 control points, then the B-spline basis of order $\frac{d z_t}{dt} = v_t(z_t), \quad t \in [0,1]$1 is constructed recursively as: $\frac{d z_t}{dt} = v_t(z_t), \quad t \in [0,1]$2 where $\frac{d z_t}{dt} = v_t(z_t), \quad t \in [0,1]$3, and $\frac{d z_t}{dt} = v_t(z_t), \quad t \in [0,1]$4 is compactly supported on $\frac{d z_t}{dt} = v_t(z_t), \quad t \in [0,1]$5. The transformation is then

$\frac{d z_t}{dt} = v_t(z_t), \quad t \in [0,1]$6

for each (active) channel in a coupling flow or as a vector path $\frac{d z_t}{dt} = v_t(z_t), \quad t \in [0,1]$7 when interpolating trajectories in CNFs.

2. Diffeomorphism and Regularity Conditions

A mapping parameterized by B-splines is a diffeomorphism if it is invertible and possesses sufficient smoothness (regularity). The fundamental sufficient condition, as established in (Hong et al., 2023), is for the transformation’s derivative to be strictly positive and bounded: $\frac{d z_t}{dt} = v_t(z_t), \quad t \in [0,1]$8 with $\frac{d z_t}{dt} = v_t(z_t), \quad t \in [0,1]$9. If, for all $v_t$0, $v_t$1 for positive constants $v_t$2, then $v_t$3 everywhere and $v_t$4 is a $v_t$5-diffeomorphism and bi-Lipschitz on $v_t$6. In practical neural parameterizations, these constraints are enforced via softplus and cumulative summation steps to guarantee monotonicity and invertibility. On periodic domains such as $v_t$7, appropriate constraints and matching of endpoint derivatives yield a periodic diffeomorphism.

The analytic inverse for cubic B-spline ($v_t$8) transforms is constructed locally by inverting a cubic equation per cell, which allows for efficient, exact inversion and evaluation of derivatives.

3. Flow Matching and B-spline Conditional Paths

In the context of flow matching for dynamical systems, diffeomorphic B-spline flows enable the interpolation of observed states at arbitrary timepoints using smooth B-spline curves. Suppose observations $v_t$9 are given at times $v_t$0; a control point solution yields $v_t$1 such that $v_t$2 and $v_t$3 (where $v_t$4 is the spline degree) (Rathod et al., 30 Jan 2026). This avoids the instability and oscillation of high-degree polynomial interpolants.

For the conditional flow, the target distribution at each $v_t$5 is modeled as

$v_t$6

with the target velocity given by $v_t$7. The regression objective minimizes the mean squared error between the neural velocity field $v_t$8 and the spline derivative $v_t$9, i.e.,

$t$0

with stochastic extensions handled by introducing a separate score network for SDE-based training.

4. Implementation and Algorithmic Aspects

Diffeomorphic B-spline flows can be implemented as CNFs with B-spline trajectory interpolants or as standard (discrete) normalizing flows with B-spline parameterizations in coupling layers. Algorithmic steps for SplineFlow (Rathod et al., 30 Jan 2026) include:

• Solving a linear system per trajectory to obtain control-point coefficients for the spline interpolant.
• Uniform sampling over the time domain and conditional Gaussian sampling from the spline-defined marginal.
• Efficient computation of spline derivatives (and, for cubic B-splines, analytic inversion).
• Regularization or architectural constraints guarantee strict monotonicity and bounded derivatives throughout training, as described in (Hong et al., 2023).
• At evaluation, the density evolution can be tracked by integrating the instantaneous divergence along the path or through direct Jacobian–trace computation.

Analytic inversion for cubic B-spline flows enables efficient sampling and log-density computation, with full inverses and higher derivatives accessible in closed form within each cell.

5. Theoretical Properties and Guarantees

The theoretical underpinnings for diffeomorphic B-spline flows include:

• Approximation error: If the ground truth $t$1, SplineFlow’s $t$2-degree spline interpolant achieves

$t$3

in contrast to $t$4 for linear paths (Rathod et al., 30 Jan 2026).

• Validity of the conditional path: The conditional formulation produces marginals matching observed data at all $t$5 and satisfies the continuity equation for the induced velocity field.
• Diffeomorphism: Enforced via bounded positive derivative constraints; $t$6 regularity is both necessary and sufficient for physical interpretability in domains such as force field modeling in molecular dynamics (Hong et al., 2023).
• Bi-Lipschitzness: The lower and upper bound constraints on the derivative guarantee quantitative stability and invertibility.
• Periodic extension: On $t$7, parameter tying yields maps that are diffeomorphisms of the circle with all derivatives up to order $t$8 matched at the boundaries.

6. Empirical Evaluation and Applications

Diffeomorphic B-spline flows have demonstrated strong empirical performance across deterministic and stochastic dynamical systems and data distributions with complex support:

• For dynamical systems, SplineFlow (Rathod et al., 30 Jan 2026) surpasses NeuralODE, LatentODE, and TrajectoryFM (linear) models in mean squared error, especially under nonlinear or irregular observation sampling regimes. In high-complexity systems (e.g., chaotic Lorenz), higher-degree B-splines recover the attractor structure lost by linear path approaches.
• On stochastic SDE inference tasks, the variance-reduced spline-based flow-matching schemes outperform static and non-spline baselines on multiple distributional metrics (Wasserstein, MMD).
• For biological trajectory data (e.g., single-cell fate inference), SplineFlow achieves superior interpolation and extrapolation accuracy versus alternative flow-matching and optimal transport approaches.
• In static distribution modeling, especially for applications demanding well-defined second-order derivatives (e.g., force computation in physical simulation with Boltzmann generators), the neural diffeomorphic non-uniform B-spline flows (Hong et al., 2023) attain negative log-likelihood and force error competitive with infinitely smooth bump-function flows, but with much faster sampling (approximately $t$9 speedup).

A summary of qualitative and quantitative findings from both frameworks is provided below:

Property / Metric	SplineFlow (Rathod et al., 30 Jan 2026)	Neural Diffeomorphic B-spline (Hong et al., 2023)
Interpolation accuracy (ODE)	SOTA for nonlinear, irregular data	Not applicable
Stability, smoothness	Arbitrary degree, non-oscillatory	$z_0 \mapsto z_t$0 (e.g. cubic, $z_0 \mapsto z_t$1)
Inverse, log-det computation	Standard CNF or explicit Jacobian	Analytic for $z_0 \mapsto z_t$2 (e.g. cubic)
Force-matching error (MD/Boltz)	Not evaluated	Matches or beats smooth flows
Sampling runtime	Simulation-free (fast)	$z_0 \mapsto z_t$3/sample

Empirical ablations confirm the importance of spline degree: for highly nonlinear or oscillatory dynamics, $z_0 \mapsto z_t$4 is essential; for smoother systems, linear paths suffice. In high-dimensional physical modeling, spline flows provide robust force fields and improved dynamic stability compared to RQ-spline or non-B-spline competitors.

7. Limitations and Outlook

Limitations of current diffeomorphic B-spline flows include:

• Occasional numerical instability in analytic inversion, primarily with cubic-root solvers under force-matching losses.
• Higher parameter and computational requirements versus traditional affine coupling flows, though substantially less than infinitely smooth non-analytic normalizing flows.
• Expressivity in high-dimensional full tensor-product B-spline flows is impractical; coupling-style architectures are typically adopted.

Despite these constraints, diffeomorphic B-spline flows constitute a flexible modeling class balancing smoothness, invertibility, analytic tractability, and empirical accuracy, with diverse applications in dynamical system inference and computational physics. Their success suggests further exploration into higher-order, domain-adapted, and structured B-spline-based transformations for both continuous and discrete-time generative modeling (Rathod et al., 30 Jan 2026, Hong et al., 2023).