Two-Parameter Flows: A Unified Perspective

• Two-parameter flows are dynamic systems defined by two independent variables, such as time, geometry, or parameter space, coordinating their evolution.
• They underpin methodologies in machine learning, fluid simulation, and discrete geometry by unifying diverse evolution laws under a common framework.
• Applications range from population dynamics inference with sampling-time flows to optimization and discrete curvature evolution demonstrating complex, scalable behavior.

Two-parameter flows are families of evolutions indexed, constrained, or generated by two independent parameters, but the expression is not used in a single standardized sense across the literature. In the works considered here, it denotes, among other things, a physical-time/sampling-time transport map for learning population dynamics from unlabeled marginals, a two-parameter family of discrete curvature flows $(\alpha,\beta)$ on triangulated manifolds, a two-way algebraic flow with two compatible multiplications, and parameter-dependent capacities or density morphisms over a two-dimensional parameter domain (Schwerdtner et al., 25 May 2026, Ge et al., 2017, Bryc et al., 2020, Allman et al., 2021).

1. Terminological scope and recurring structures

In machine learning and scientific inference, “two-parameter” often refers to explicit continuous control variables. Factorizable Normalizing Flows treat the target density as $p(y\mid x,\nu)$ with $\nu=(\nu_1,\dots,\nu_K)$, and in the two-parameter specialization the affine autoregressive coefficients are sums of parameter-specific linear and quadratic responses, optionally augmented by bilinear interaction terms $\nu_1\nu_2\,\phi_j^{12}$ and $\nu_1\nu_2\,\psi_j^{12}$ (Valsecchi et al., 29 Jun 2026). In parameter-conditioned fluid simulation, the conditioning vector itself can be two-dimensional, as in $c=[\Omega,\mathrm{Re}]^T$, and the problem is to learn the conditional trajectory law $p_\theta(x_{1:T}\mid c)$ over a two-parameter flow family rather than at a single operating point (Morton et al., 2019).

In optimization and graph algorithms, the same phrase can denote dependence on a two-dimensional parameter plane. For source-sink monotone min cut, arc capacities take the affine form $u_{ij}(\lambda,\mu)=a_{ij}\lambda+b_{ij}\mu+c_{ij}$, and the parameter space is ordered by the product order on $(\lambda,\mu)$ (Allman et al., 2021). In stochastic-flow theory, by contrast, the two-parameter object is usually the time pair $(s,t)$: stochastic flows of mappings and kernels satisfy cocycle identities such as $p(y\mid x,\nu)$0 and $p(y\mid x,\nu)$1 (Jan et al., 2011).

This suggests that the common feature is not a single canonical formula, but the presence of two independent directions of variation. Depending on the field, these may be two control parameters, two time arguments, two geometric exponents, or two coupled evolution variables.

2. Physical-time and sampling-time flows in population-dynamics inference

A particularly explicit modern formulation appears in two-parameter flows for learning population dynamics from unlabeled time marginals. The data are independent samples $p(y\mid x,\nu)$2, with no trajectory correspondences across times, and the target is a population-level velocity field $p(y\mid x,\nu)$3 satisfying the continuity equation

$p(y\mid x,\nu)$4

The difficulty is nonuniqueness: if $p(y\mid x,\nu)$5 is admissible and $p(y\mid x,\nu)$6, then $p(y\mid x,\nu)$7 is also admissible. The proposed resolution is to introduce an auxiliary sampling-time variable $p(y\mid x,\nu)$8 and learn only transports from a base distribution $p(y\mid x,\nu)$9 to each marginal $\nu=(\nu_1,\dots,\nu_K)$0 (Schwerdtner et al., 25 May 2026).

The construction uses a sampling-time velocity $\nu=(\nu_1,\dots,\nu_K)$1 and its flow

$\nu=(\nu_1,\dots,\nu_K)$2

which induces

$\nu=(\nu_1,\dots,\nu_K)$3

The second parameter is physical time $\nu=(\nu_1,\dots,\nu_K)$4. The induced physics-time velocity is defined by

$\nu=(\nu_1,\dots,\nu_K)$5

while the sampling-time velocity satisfies

$\nu=(\nu_1,\dots,\nu_K)$6

Because $\nu=(\nu_1,\dots,\nu_K)$7 is assumed $\nu=(\nu_1,\dots,\nu_K)$8, mixed derivatives commute, yielding the compatibility condition

$\nu=(\nu_1,\dots,\nu_K)$9

The resulting $\nu_1\nu_2\,\phi_j^{12}$0 is unique once $\nu_1\nu_2\,\phi_j^{12}$1 is fixed and regular enough. The paper proves that if $\nu_1\nu_2\,\phi_j^{12}$2 and the $\nu_1\nu_2\,\phi_j^{12}$3-flow has lifetime one, then $\nu_1\nu_2\,\phi_j^{12}$4 is a $\nu_1\nu_2\,\phi_j^{12}$5-diffeomorphism, $\nu_1\nu_2\,\phi_j^{12}$6 is uniquely defined by $\nu_1\nu_2\,\phi_j^{12}$7, and $\nu_1\nu_2\,\phi_j^{12}$8 satisfies the continuity equation in the weak sense. It also gives the explicit formula

$\nu_1\nu_2\,\phi_j^{12}$9

with $\nu_1\nu_2\,\psi_j^{12}$0 the Jacobian of $\nu_1\nu_2\,\psi_j^{12}$1 with respect to $\nu_1\nu_2\,\psi_j^{12}$2. A further $\nu_1\nu_2\,\psi_j^{12}$3-bound shows that regularity of $\nu_1\nu_2\,\psi_j^{12}$4 is inherited from regularity of $\nu_1\nu_2\,\psi_j^{12}$5.

Operationally, the method first trains $\nu_1\nu_2\,\psi_j^{12}$6 by conditional flow matching, using stochastic interpolants $\nu_1\nu_2\,\psi_j^{12}$7. It then samples $\nu_1\nu_2\,\psi_j^{12}$8, constructs synthetic trajectories $\nu_1\nu_2\,\psi_j^{12}$9, and regresses a model $c=[\Omega,\mathrm{Re}]^T$0 to their finite-difference derivatives. The stated advantage is that this avoids per-step optimal-transport couplings, scales through standard conditional flow matching machinery, and permits admissible non-gradient dynamics, including rotational or circulating phenomena.

3. Two-parameter density morphing and parameter-conditioned simulation

A related but distinct use of two-parameter flows appears in parameter-dependent density morphing. Factorizable Normalizing Flows represent $c=[\Omega,\mathrm{Re}]^T$1 as a fixed reference density $c=[\Omega,\mathrm{Re}]^T$2 composed with an invertible map $c=[\Omega,\mathrm{Re}]^T$3, so that

$c=[\Omega,\mathrm{Re}]^T$4

In the two-parameter case, the affine autoregressive coefficients are

$c=[\Omega,\mathrm{Re}]^T$5

with optional interaction terms $c=[\Omega,\mathrm{Re}]^T$6 and $c=[\Omega,\mathrm{Re}]^T$7. In the factorized setting, the model is trained only on the four single-nuisance templates $c=[\Omega,\mathrm{Re}]^T$8, never on joint settings such as $c=[\Omega,\mathrm{Re}]^T$9. Near the nominal point and along the training axes, the KL error is reported as a few $p_\theta(x_{1:T}\mid c)$0 nats/event; it exceeds $p_\theta(x_{1:T}\mid c)$1 nats/event near the corners when both nuisances are large, and adding pairwise cross terms suppresses the residual across the plane to below $p_\theta(x_{1:T}\mid c)$2 nats/event (Valsecchi et al., 29 Jun 2026).

A second nearby formulation is parameter-conditioned sequential generative modeling of fluid flows. There the object is a conditional latent-variable model $p_\theta(x_{1:T}\mid c)$3, and the main two-dimensional experiment uses

$p_\theta(x_{1:T}\mid c)$4

for counter-rotating cylinders. The model is trained on CFD data over $p_\theta(x_{1:T}\mid c)$5 and $p_\theta(x_{1:T}\mid c)$6, and is then evaluated over a dense two-parameter sweep. Reported results include qualitative accuracy at an untrained setting $p_\theta(x_{1:T}\mid c)$7, a largest relative error of $p_\theta(x_{1:T}\mid c)$8 across the tested local and global flow quantities, nearly 13,000 learned simulations in a little under 10 hours, and an overall speedup of approximately $p_\theta(x_{1:T}\mid c)$9 relative to CFD (Morton et al., 2019).

These models treat two-parameter flows as families of densities or trajectories over a two-dimensional conditioning domain. This suggests a broad machine-learning usage in which “flow” refers less to a unique physical law than to a scalable generative representation of how distributions deform across parameter space.

4. Constitutive uses in continuum mechanics

In dense granular rheology, the phrase “two-parameter flows” arises in a constitutive sense: whether a local friction law should depend only on inertial number $u_{ij}(\lambda,\mu)=a_{ij}\lambda+b_{ij}\mu+c_{ij}$0 or on $u_{ij}(\lambda,\mu)=a_{ij}\lambda+b_{ij}\mu+c_{ij}$1 plus a second parameter describing flow type. The study compares three steady two-dimensional granular geometries—rough incline flow, incline flow with a circular intruder, and a discharging planar silo—and defines the effective friction coefficient

$u_{ij}(\lambda,\mu)=a_{ij}\lambda+b_{ij}\mu+c_{ij}$2

To classify local flow type, it introduces $u_{ij}(\lambda,\mu)=a_{ij}\lambda+b_{ij}\mu+c_{ij}$3, where in linearized isochoric form

$u_{ij}(\lambda,\mu)=a_{ij}\lambda+b_{ij}\mu+c_{ij}$4

so that $u_{ij}(\lambda,\mu)=a_{ij}\lambda+b_{ij}\mu+c_{ij}$5 is simple shear, $u_{ij}(\lambda,\mu)=a_{ij}\lambda+b_{ij}\mu+c_{ij}$6 is pure extension, and $u_{ij}(\lambda,\mu)=a_{ij}\lambda+b_{ij}\mu+c_{ij}$7 is solid-body rotation (Bhateja et al., 2017).

The key result is negative for a simple two-parameter constitutive law. Each geometry separately exhibits a monotonic $u_{ij}(\lambda,\mu)=a_{ij}\lambda+b_{ij}\mu+c_{ij}$8-type scaling well fit by the Jop form, but the fitted curves are not universal across geometries. The inclined-flow data reproduce the familiar unidirectional-shear behavior, the silo data collapse across different outlet sizes but lie on a different $u_{ij}(\lambda,\mu)=a_{ij}\lambda+b_{ij}\mu+c_{ij}$9 curve, and the incline-plus-intruder data lie between them. When $(\lambda,\mu)$0 is plotted against $(\lambda,\mu)$1, the paper reports that there is no apparent correlation. It explicitly states that “There does not appear to be any correlation indicating that a modification of Eq.~(\ref{eqn:mulaw}) incorporating the flow parameter may not be a useful approach.”

The positive result is instead fluctuation-based. Using the Zhang–Kamrin scaling,

$(\lambda,\mu)$2

the scaled granular fluidity collapses “reasonably well” across the incline, intruder, and silo geometries when plotted against solid fraction $(\lambda,\mu)$3. At the same time, $(\lambda,\mu)$4 itself collapses well against $(\lambda,\mu)$5 across all geometries. In this usage, “two-parameter flows” therefore marks a failed constitutive hypothesis of the form $(\lambda,\mu)$6, and the paper’s evidence points instead toward packing fraction and fluctuation-aware fluidity as the more effective organizing variables.

5. Algebraic, stochastic, and translation-equation formalisms

In algebraic probability, two-way flows arise in double near algebras, where a single linear space carries two associative left-linear multiplications. A family $(\lambda,\mu)$7 is a two-way flow if it is invertible with respect to both multiplications and satisfies two composition laws together with a structure condition. The paper’s main representation theorem states that every such flow is generated by a single element $(\lambda,\mu)$8 through

$(\lambda,\mu)$9

with

$(s,t)$0

The converse is exact once $(s,t)$1 satisfies an additional compatibility condition defining a flow generator (Bryc et al., 2020).

In stochastic analysis, the canonical two-parameter object is indexed by start and end times. For Brownian flows on $(s,t)$2, the paper studies a sign-dependent SDE driven by two independent white noises and distinguishes a stochastic flow of mappings $(s,t)$3 from a stochastic flow of kernels $(s,t)$4. It proves that the two-noise equation has a unique coalescing mapping flow $(s,t)$5, that the filtered object

$(s,t)$6

is the unique diffusive Wiener solution of the corresponding kernel equation, and that a second coalescing flow $(s,t)$7 has the same filtered kernel law despite not being a Wiener solution. The map/kernel distinction is thus intrinsic to the two-parameter framework (Jan et al., 2011).

A third mathematical usage appears in the projective translation equation. There a plane flow is written

$(s,t)$8

and the additive-time law $(s,t)$9 is reduced to the one-step functional equation

$p(y\mid x,\nu)$00

The classification over $p(y\mid x,\nu)$01 is complete up to conjugation by $p(y\mid x,\nu)$02-homogenic birational maps: there are a zero flow, two singular flows, an identity flow, and one non-singular canonical flow for each level $p(y\mid x,\nu)$03, with representative

$p(y\mid x,\nu)$04

Here the two parameters are the spatial variable $p(y\mid x,\nu)$05 and the additive time $p(y\mid x,\nu)$06, and the theory is organized through birational equivalence, homogeneous vector fields, and algebraic orbit invariants (Alkauskas, 2012).

6. Discrete geometry and combinatorial optimization

In discrete differential geometry, two-parameter flows are explicit families indexed by two exponents. On a closed triangulated manifold $p(y\mid x,\nu)$07 with $p(y\mid x,\nu)$08 or $p(y\mid x,\nu)$09, the unified $p(y\mid x,\nu)$10-flow is

$p(y\mid x,\nu)$11

where $p(y\mid x,\nu)$12 is the discrete $p(y\mid x,\nu)$13-curvature and

$p(y\mid x,\nu)$14

This framework unifies several earlier flows, including the Chow–Luo combinatorial Ricci flow at $p(y\mid x,\nu)$15, various 2D and 3D $p(y\mid x,\nu)$16-flows, and normalized Yamabe-type evolutions. If $p(y\mid x,\nu)$17, then either $p(y\mid x,\nu)$18 or $p(y\mid x,\nu)$19 is invariant. Critical points are exactly constant $p(y\mid x,\nu)$20-curvature metrics, and in dimension $p(y\mid x,\nu)$21 local convergence to a constant $p(y\mid x,\nu)$22-curvature metric $p(y\mid x,\nu)$23 holds under

$p(y\mid x,\nu)$24

in particular when $p(y\mid x,\nu)$25 (Ge et al., 2017).

In combinatorial optimization, two-parameter dependence can have the opposite effect: it can destroy the low-complexity behavior familiar from one-parameter problems. For source-sink monotone min cut with affine capacities

$p(y\mid x,\nu)$26

nestedness of minimum cuts still follows from the product order on $p(y\mid x,\nu)$27, but the number of distinct min-cut cells can be exponential. The main theorem states that there exist two-parameter instances in which all $p(y\mid x,\nu)$28 $p(y\mid x,\nu)$29-$p(y\mid x,\nu)$30 cuts occur as unique minimum cuts for some parameter values. Along any single monotone northeast path there are still at most $p(y\mid x,\nu)$31 cut changes, but globally the two-dimensional parameter plane admits exponentially many regions (Allman et al., 2021).

Taken together, these results show that the addition of a second parameter can either unify a large class of flows, as in discrete curvature evolution, or fundamentally increase combinatorial complexity, as in parametric min cut.

7. Conceptual synthesis and terminological boundaries

Across these literatures, two-parameter flows are best understood as structured evolutions in which two independent variables must be coordinated. In population-dynamics inference, the variables are sampling time $p(y\mid x,\nu)$32 and physics time $p(y\mid x,\nu)$33. In density morphing and surrogate simulation, they are external conditioning parameters such as $p(y\mid x,\nu)$34 or $p(y\mid x,\nu)$35. In stochastic-flow theory, they are start and end times $p(y\mid x,\nu)$36. In discrete geometry, they are the curvature and metric exponents $p(y\mid x,\nu)$37. In rheology, the phrase marks the constitutive question of whether inertial number $p(y\mid x,\nu)$38 must be supplemented by a flow-type descriptor such as $p(y\mid x,\nu)$39.

A common misconception is that any technical phrase containing “two” refers to this same concept. The accretion literature provides two nearby counterexamples. “Two temperature accretion flows” describe optically thin viscous disks with separate ion and electron temperatures, Coulomb coupling $p(y\mid x,\nu)$40, and radiative cooling $p(y\mid x,\nu)$41; the “two” refers to species temperatures, not to a two-parameter flow law (Mukhopadhyay, 2011). Similarly, “Two Component Advective Flow” denotes a Keplerian disk plus a sub-Keplerian halo in black-hole accretion, produced numerically through vertically stratified viscosity and cooling; again, the terminology concerns component structure rather than a generic two-parameter formalism (Giri et al., 2012).

The literature therefore does not support a single universal definition. What it does support is a family resemblance: two-parameter flows arise whenever one must preserve consistency across two independent axes of evolution or deformation, whether those axes are temporal, geometric, constitutive, probabilistic, or parametric.