Conditioned Flows: Conditional Generative Models

• Conditioned flows are invertible generative models that condition transformation on external variables to model complex, structured data with tractable likelihoods.
• They employ diverse architectures like ConvLSTMs, permutation-invariant networks, and continuous time flows to integrate discrete, spatial, and temporal information.
• Training driven by maximum likelihood yields efficient sampling and calibrated uncertainty estimates, facilitating robust application in climate, fluid dynamics, and trajectory forecasting.

Conditioned flows are a class of invertible generative models wherein the distributional transformation is parametrically dependent on external conditioning variables, enabling explicit modeling of conditional densities $p(x \mid c)$. These methods extend the foundational normalizing flow paradigm by incorporating context-dependent conditioning into the flow's construction—permitting structured uncertainty quantification, enhanced control over output attributes, and tractable likelihoods in diverse structured data regimes. Architectures and techniques for conditioned flows exploit conditioning across discrete, continuous, spatial, temporal, and set-structured data, facilitating a wide spectrum of scientific, engineering, and generative modeling applications.

1. Mathematical Formulation of Conditioned Flows

Conditioned flows generalize standard normalizing flows by introducing a dependence on an external condition $c$ (e.g., temporal context, physical parameter, scene embedding). The defining change-of-variables relation becomes: $$p(x \mid c) = p_0\left(f(x; c)\right) \cdot \left| \det \frac{\partial f(x; c)}{\partial x} \right|$$ where $f$ is an invertible mapping depending on $c$ and $p_0$ is a base density (commonly standard normal). In continuous-time (CNF) variants, $f$ is defined via ODEs with vector fields explicitly conditioned on $c$, yielding

$\frac{dX(t)}{dt} = f\bigl(t, X(t), c\bigr)$

and the conditional log-density is given by integrating the divergence of $f$ along the flow path.

The conditioning enters at multiple levels:

• By concatenating $c$0 (or an encoding $c$1) into every flow layer, either as an input to the conditioner network in (affine or spline) coupling layers,
• By modulating flow parameters (e.g., batchnorm, convolution kernels) with $c$2,
• Via explicit conditioning of the prior in latent space: $c$3, commonly implemented as a diagonal Gaussian with parameters predicted from $c$4,
• In continuous (amortized) settings via vector-field conditioning for conditional density families.

This structure allows flows to model families of distributions indexed by $c$5, with tractable sampling and likelihoods for inference, model selection, and downstream risk quantification (Winkler et al., 2023, Generale et al., 2024, Rasul et al., 2020).

2. Core Architectures and Conditioning Strategies

Conditioned flow architectures are generally agnostic to the coupling layers' detailed form (RealNVP, Glow, rational-quadratic splines, continuous flows, etc.) but require context injection at each layer. Representative mechanisms include:

• Spatial/Temporal Convolutional Injection: For spatio-temporal data, context $c$6 is encoded via convolutional LSTMs (ConvLSTM) or masked convolutions, producing a high-dimensional hidden state $c$7 that is concatenated or injected (as conditional batchnorm or additional input channels) into each flow coupling layer. This is standard in spatio-temporal prediction settings such as climate emulation (Winkler et al., 2023) and trajectory forecasting (Zand et al., 2021). In fully autoregressive decompositions, masked convolutions (ARNs/LMConv) permit flexible causal ordering.
• Parameter and Physical Context Conditioning: For parameterized simulation tasks, $c$8 can be a vector of physical parameters (e.g., Reynolds number, geometry) concatenated to the feature extractor or directly used in FiLM-like affine modulators in the prior and posterior networks (Morton et al., 2019).
• Latent Structural Conditioning: When modeling set- or permutation-invariant data, permutation-equivariant network components (e.g., sum of per-element $c$9 and pairwise $$p(x \mid c) = p_0\left(f(x; c)\right) \cdot \left| \det \frac{\partial f(x; c)}{\partial x} \right|$$0 interactions) guarantee the flow respects the exchangeability of $$p(x \mid c) = p_0\left(f(x; c)\right) \cdot \left| \det \frac{\partial f(x; c)}{\partial x} \right|$$1 given $$p(x \mid c) = p_0\left(f(x; c)\right) \cdot \left| \det \frac{\partial f(x; c)}{\partial x} \right|$$2 (Zwartsenberg et al., 2022).
• Continuous-Conditioning (Amortized) Flows: To allow models to function across a continuous space of $$p(x \mid c) = p_0\left(f(x; c)\right) \cdot \left| \det \frac{\partial f(x; c)}{\partial x} \right|$$3, Conditional Variable Flow Matching (CVFM) and related methods learn vector fields $$p(x \mid c) = p_0\left(f(x; c)\right) \cdot \left| \det \frac{\partial f(x; c)}{\partial x} \right|$$4 amortized over $$p(x \mid c) = p_0\left(f(x; c)\right) \cdot \left| \det \frac{\partial f(x; c)}{\partial x} \right|$$5. This enables predictions for unpaired or previously unseen $$p(x \mid c) = p_0\left(f(x; c)\right) \cdot \left| \det \frac{\partial f(x; c)}{\partial x} \right|$$6 and leverages conditional optimal transport to ensure correct interpolations (Generale et al., 2024).

3. Training Objectives and Probabilistic Inference

The predominant training objective for conditioned flows is maximum likelihood, frequently with exact (tractable) likelihood for both training and calibration: $$p(x \mid c) = p_0\left(f(x; c)\right) \cdot \left| \det \frac{\partial f(x; c)}{\partial x} \right|$$7 For autoregressive spatio-temporal models, likelihood is decomposed across time, propagating the context sequentially. In settings utilizing variational methods, e.g., parameter-conditioned sequential VAEs or CF-VAE, the negative log-likelihood is incorporated into the evidence lower bound (ELBO), often with additional regularization (mutual-information, posterior regularization) to ensure the capacity and identifiability of the conditional latent variables (Bhattacharyya et al., 2019, Morton et al., 2019).

Auxiliary terms are introduced for specialized constraints, e.g., mass-constraint penalties in conditional jet generation, semigroup consistency for single-step transport fields in motion, or conditional OT/Wasserstein losses when learning families of conditional densities (Käch et al., 2022, Bella et al., 28 Apr 2026, Generale et al., 2024).

At inference, sampling conditioned on $$p(x \mid c) = p_0\left(f(x; c)\right) \cdot \left| \det \frac{\partial f(x; c)}{\partial x} \right|$$8 is achieved efficiently via ancestral passes (e.g., sampling multiscale latent slices and running inverted flow steps), and likelihoods are computed exactly for principled calibration and integration into uncertainty quantification pipelines.

4. Application Domains and Empirical Performance

Conditioned flows have proven effective across numerous structured data regimes:

• Spatio-temporal climate variable emulation: Conditioned spatio-temporal flows ("ST-Flow") surpass deterministic and stochastic baselines on long-range extrapolation of temperature and geopotential variables, uniquely enabling stable rollouts beyond training windows with calibrated uncertainty estimates (Winkler et al., 2023).
• Computational fluid dynamics: Parameter-conditioned flows enable rapid, accurate generation of 2D and 3D unsteady flows across varying Reynolds numbers and system parameters, facilitating 120-fold accelerations over CFD solvers while preserving global and local physical statistics (Morton et al., 2019).
• Trajectory and motion prediction: Time-conditioned or scene-conditioned flows, e.g., MotionFlow and FlowS, can efficiently generate multimodal agent forecasts, achieving top performance on benchmarks such as Waymo Open Motion, NBA player trajectories, and human pose sequences (Zand et al., 2021, Bella et al., 28 Apr 2026).
• Set-structured and permutation-invariant data: Conditional permutation invariant flows model traffic scenes and object configurations with explicit exchangeability, delivering state-of-the-art log-likelihoods and realistic sample generation (Zwartsenberg et al., 2022).
• Attribute-conditioned generative models: Conditional continuous flows, e.g., StyleFlow, enable precise manipulation of GAN latent spaces conditioned on semantic attributes, outperforming linear and unconditioned methods in attribute-controlled image editing (Abdal et al., 2020).
• Physics and geometry-constrained domains: Specialized boundary-conditioned flows (Waveflow) construct valid antisymmetric many-body fermion wavefunctions, resolving topological incompatibilities and enabling accurate ground-state recovery in quantum systems (Thiede et al., 2022).

5. Extensions: Advanced Conditioning, Structure, and Kernel Theory

Recent advances emphasize both the necessity and theoretical subtlety of conditioning in flow-based models:

• Support-Conditioned and Kernel-Smoothed Flows: Under finite support and Gaussian OT, conditioning reduces to a Nadaraya–Watson kernel smoother, interpretable as single-head cross-attention with dynamic bandwidth tied to the flow time. This yields explicit predictions for failure modes (high-dimensional collapse, geometry mismatch, insufficient support), with implications for reference-guided and cross-attention generative models (Smola, 13 May 2026).
• Amortized Conditional Optimal Transport: CVFM provides scalability in learning conditional densities over continuous $$p(x \mid c) = p_0\left(f(x; c)\right) \cdot \left| \det \frac{\partial f(x; c)}{\partial x} \right|$$9, leveraging Sinkhorn-regularized couplings and reweighted kernel matching to enable well-behaved, stable generalization across unpaired datasets and high-dimensional settings (Generale et al., 2024).
• Explicit Constructive Theory: For any target conditional mapping (triangular transport), constructive theory guarantees explicit flow realizations using ODEs governed by (possibly width-1) perceptrons with piecewise-constant weights, via polar factorization and measure-preserving shear flows—even providing precise scaling in terms of approximation error and ambient dimension (Geshkovski et al., 9 Feb 2026).

6. Empirical Insights, Limitations, and Future Prospects

Empirical studies demonstrate that conditioned flows offer:

• Exact likelihoods for principled model comparison and downstream probabilistic integration,
• Efficient sampling and rollouts suitable for real-time applications (climate exploration, motion prediction, interactive urban acoustics),
• High-fidelity uncertainty quantification, with latent variable slicing and calibrated generative process,
• State-of-the-art results in a wide range of scientific and machine learning tasks (Winkler et al., 2023, Morton et al., 2019, Zand et al., 2021, Zwartsenberg et al., 2022, Käch et al., 2022, Bella et al., 28 Apr 2026).

However, performance is sensitive to architectural alignment with data symmetry and geometry (permutation equivariance, kernel adaptivity), available support, and conditioning bandwidth. High-dimensional data introduces risk of nearest-neighbor collapse or variance blowup when using isotropic kernels; adapting conditioning mechanisms (e.g., multi-head projections, geometric encoders) is crucial (Smola, 13 May 2026). For tasks involving irregular grids, hybrid discrete-continuous structures, or strong physical constraints, extensions such as physics-informed priors, graph convolutions, or manifold-aware layers are suggested (Winkler et al., 2023, Sengupta et al., 2023).

Ongoing research focuses on improving the statistical efficiency of conditioning, alleviating sampling bottlenecks for variable-size and set-valued data, jointly modeling conditioning-variable distributions, and integrating flow-based approaches with optimal transport, kernel, and cross-attention theory for robust, interpretable conditional generative modeling.

7. Summary Table: Canonical Conditioned Flow Architectures

Domain	Conditioning Form	Core Flow Mechanism	Key Reference
Spatio-temporal climate	Temporal ConvLSTM	Affine coupling, multiscale	(Winkler et al., 2023)
Fluid simulations	Parameter vector $f$0	Autoregressive latent flow	(Morton et al., 2019)
Permutation-invariant sets	Scene image or map	CNF with equivariant ODE	(Zwartsenberg et al., 2022)
Motion/trajectory	Scene & agent context	Conditional flow matching	(Bella et al., 28 Apr 2026)
GAN latent editing	Attribute vector $f$1	CNF with ConcatSquash	(Abdal et al., 2020)
Particle physics jets	Mass & count	Spline coupling flows	(Käch et al., 2022)
Kernel-constrained flows	Support set $f$2	NW kernel smoothing (attention)	(Smola, 13 May 2026)
General conditional FM	Continuous $f$3	OT-matched vector fields	(Generale et al., 2024)

Conditioned flows constitute a fundamental toolset for conditional generative modeling, offering tractable, expressive, and scientifically grounded approaches to learning complex structured conditional distributions across broad application domains.