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The Principles of Diffusion Models

Chieh-Hsin Lai, Yang Song, Dongjun Kim, Yuki Mitsufuji, Stefano Ermon

arXiv:2510.21890v3cs.LGcs.AIcs.GR

TL;DR

Diffusion generation can be computationally intensive because differential-equation solvers are iterative and often require many steps. The book organizes reverse modeling through variational, score-based, and flow-based perspectives, while discussing accelerated solvers and controllable generation.

  • Problem

    Diffusion sampling is computationally intensive because numerical solvers iteratively approximate differential-equation trajectories, often requiring many steps.

  • Method

    The book frames reverse modeling through variational, score-based, and flow-based perspectives and uses the PF-ODE formulation to motivate accelerated numerical solvers.

  • Results

    The three-way viewpoint carries over to discrete diffusion models as three ways to identify the unknown object needed to reverse the probability path.

  • Takeaways & Limitations

    The PF-ODE formulation enables use of the extensive literature on accelerated numerical solvers for diffusion sampling.

  • Takeaways & Limitations

    Classifier guidance requires training a classifier to operate across all noise levels, adding a training-cost constraint.

Abstract

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This book presents the core principles that have guided the development of diffusion models, tracing their origins and showing how diverse formulations arise from shared mathematical ideas. Diffusion modeling starts by defining a forward process that gradually corrupts data into noise, linking the data distribution to a simple prior through a continuum of intermediate distributions. The goal is to learn a reverse process that transforms noise back into data while recovering the same intermediates. We describe three complementary views. The variational view, inspired by variational autoencoders, sees diffusion as learning to remove noise step by step. The score-based view, rooted in energy-based modeling, learns the gradient of the evolving data distribution, indicating how to nudge samples toward more likely regions. The flow-based view, related to normalizing flows, treats generation as following a smooth path that moves samples from noise to data under a learned velocity field. These perspectives share a common backbone: a time-dependent velocity field whose flow transports a simple prior to the data. Sampling then amounts to solving a differential equation that evolves noise into data along a continuous trajectory. On this foundation, the book discusses guidance for controllable generation, efficient numerical solvers, and diffusion-motivated flow-map models that learn direct mappings between arbitrary times. It provides a conceptual and mathematically grounded understanding of diffusion models for readers with basic deep-learning knowledge. Supplementary materials for the book are available at the book website: https://the-principles-of-diffusion-models.github.io/

C Behind the Scenes of Diffusion Models:

The book includes supporting material on learning fast diffusion-based generators and an optional treatment of Elucidating Diffusion Models. The listed contributors include Chieh-Hsin Lai, Yang Song, Dongjun Kim, Yuki Mitsufuji, and Stefano Ermon.

  • The supplementary contents include a theoretical supplement on learning fast diffusion-based generators and an optional EDM section.
  • The listed authors are Chieh-Hsin Lai, Yang Song, Dongjun Kim, Yuki Mitsufuji, and Stefano Ermon.

Preface and Roadmap

The book develops a principled, durable framework for understanding diffusion models, tracing their foundations across variational, score-based, and flow-based perspectives. It also covers efficient generation, broader state spaces, and the scope and intended audience of the treatment.

  • Scope and Goals: The treatment develops essential concepts and formulations while establishing a stable foundation rather than surveying every variant or application.
  • Audience: The intended readers are researchers, graduate students, and practitioners with basic deep-learning knowledge who want a principled understanding of diffusion models.
  • Preface and Roadmap: The book traces diffusion models to three perspectives: variational, score-based, and flow-based methods.
  • Core Perspectives on Diffusion Models: Diffusion generation is presented as a gradual transformation from noise to data through forward corruption and reverse denoising.
  • Learning Fast Generative Models: The book examines fast generators, including distillation and flow-map models, that approximate diffusion with far fewer integration steps, sometimes only a few or one.
  • Beyond Diffusion on Continuous State Spaces: Its final chapter extends the three perspectives beyond continuous data to discrete objects such as text and protein sequences.

Notations

The notation section establishes conventions for vectors, matrices, derivatives, distributions, expectations, time indices, and Gaussian perturbations used throughout the book.

  • The notation defines common linear-algebra objects, including vectors, matrices, transposes, traces, identity matrices, and diagonal matrices.
  • It distinguishes learnable parameters from trained parameters fixed during inference.
  • Gradients, Jacobians, divergences, Hessians, integrals, and density notation are specified for later mathematical expressions.
  • The conventions define distributions, expectations, conditional expectations, and point probabilities across continuous and discrete settings.
  • Implicit time indices allow expressions such as E[x_s|x_t] and p(x_s|x_t) to omit time variables when context makes them clear.
  • Gaussian perturbations use x_t = α_t x_0 + σ_t ε, with x_0 drawn from p_data and ε independent of x_0; equality in distribution denotes the same probability law.

Deep Generative Modeling

Deep generative modeling learns distributions from finite samples so models can generate realistic and controllable new examples. The chapter frames training as discrepancy minimization and connects major model families through their objectives and mechanisms.

  • Chapter Overview: The chapter builds a conceptual and mathematical foundation for understanding how major generative-model families differ in learning and sampling.
  • Deep generative models learn an approximate distribution p_ϕ from examples drawn from an unknown data distribution p_data.
  • Their goals are realistic generation of novel samples and fine-grained, interpretable control over generation.
  • Because p_data is unknown, discrepancies are estimated from finite i.i.d. samples, and sufficient model capacity can yield close approximation.
  • Maximum Likelihood: Maximum likelihood trains models by assigning high density to observed data, equivalently minimizing the forward KL divergence when the model density is evaluable.
  • KL Divergences: Forward KL penalizes models that completely miss regions with data probability, a behavior known as mode covering.
  • Generative Model Families: GANs implicitly minimize the Jensen–Shannon divergence, while f-GANs place adversarial training within a broader f-divergence framework.

Foundations and Perspectives

Diffusion models define a corruption path from clean data to noise, then learn repeated predictions that move fresh noise back toward data. Variational, score-based, and flow-based perspectives organize this shared process around different modeling interpretations.

  • Forward diffusion: Diffusion defines noisy observations by mixing clean data with Gaussian noise at a chosen noise level.The coefficients α_t and σ_t control the relative contributions of clean signal and noise.
  • Training: Training samples a clean datum, noise level, and noise draw, then trains a network to predict the clean signal from only the corrupted observation and noise level.The prediction is scored with an ordinary squared-error loss against the known clean target.
  • Prediction: Squared-error training makes the optimal clean-data prediction E[x0|xt,t], averaging over plausible clean samples when observations are ambiguous.Ambiguity increases at higher noise levels, so predictions can sharpen as sampling moves toward lower noise.
  • Generation: The learned network generates samples by starting from fresh prior noise and repeatedly stepping from high noise toward lower noise levels.Each update re-mixes predicted signal and noise using coefficients for a slightly lower noise level.
  • Generation: Sampling is a sequence of revised predictions rather than recovery of one hidden corruption history.The network receives a new, less noisy state after each step and typically changes its prediction.

Variational Perspective: From VAEs to DDPMs

The variational perspective connects VAEs and DDPMs through latent-variable modeling and tractable objectives. DDPMs fix the forward encoder as progressive noising and learn its reverse through successive denoising steps.

  • DDPMs: DDPMs extend the hierarchical latent-variable view by fixing a forward noising encoder and learning a reverse generative trajectory through many denoising layers.The Markov and Gaussian structure makes reverse conditionals tractable and enables a tractable training objective.
  • VAEs: VAEs introduce latent variables with a probabilistic encoder qθ(z|x) and decoder pϕ(x|z), replacing unstructured autoencoder latents with a generative model.A simple prior supports sampling, while decoding maps latent representations back to observed data.
  • ELBO: VAEs optimize the ELBO because direct marginal-likelihood maximization and exact posterior inference are generally intractable for expressive nonlinear decoders.The encoder qθ(z|x) serves as a tractable approximation to the posterior.
  • ELBO: The ELBO combines reconstruction and latent-KL terms, balancing accurate recovery of x with a latent distribution close to a simple Gaussian prior.This trade-off supports both faithful reconstructions and coherent sampling.
  • ELBO: Maximizing the ELBO reduces inference error, the non-negative gap between the true log-likelihood and the ELBO.The joint-KL decomposition separates modeling error from the mismatch between approximate and true posteriors.
  • DDPMs: By fixing the encoder and concentrating learning on gradual reversal, DDPMs achieve stability and expressive power.The book presents this as a significant improvement in generated-output quality.

Question 2.2.1

The book develops unified mathematical connections among variational, score-based, and continuous-time diffusion formulations. It shows that denoising, score learning, and generation through differential equations are closely related views of the same modeling principles.

  • Variational perspective: The variational view makes DDPM training tractable by converting per-step KL terms into supervised denoising regressions under a fixed linear-Gaussian forward process.The forward noising chain acts as a fixed encoder, while closed-form conditional targets support efficient sampling of training objectives.
  • Score-based perspective: Denoising Score Matching provides a principled and scalable way to learn score functions, whose optimum recovers ∇_x log p_σ(x) at each noise level.Adding Gaussian noise gives the perturbed distribution full support, making its score well-defined everywhere.
  • Score-based perspective: Tweedie’s formula links score prediction with denoising by expressing the clean-sample estimate from a noisy observation through the score function.A gradient-ascent step with step size σ^2 yields the conditional average clean signal.
  • Score-based perspective: Noise prediction in DDPMs is equivalent, up to the known scale factor 1/σ_i, to predicting the score at noise level i.The NCSN optimum is the true score, while the DDPM optimum predicts the conditional noise expectation.
  • Continuous-time unification: Continuous-time diffusion unifies DDPMs and NCSNs as discretizations of an SDE, while generation becomes solving an SDE or ODE numerically.This formulation supports advanced numerical methods for improving sampling efficiency and accuracy.

Question 5.1.1

Flow Matching extends diffusion's conditional-to-marginal learning principle to transport between arbitrary source and target distributions. Its conditional velocity targets recover the marginal field, while Reflow can progressively straighten trajectories for easier numerical integration.

  • Flow Matching provides simulation-free training and learns continuous transport between arbitrary endpoint distributions.The framework defines conditional paths and velocities whose marginal flow connects psrc and ptgt.
  • Diffusion and Flow Matching both replace an intractable marginal target with tractable conditional targets, learning either the score or velocity field.Score-based diffusion learns conditional scores before constructing the PF-ODE velocity; Flow Matching applies the principle directly to velocity.
  • Averaging conditional velocity targets over training examples recovers the desired marginal velocity field.When conditional fields induce their conditional density paths, the resulting marginal field induces the corresponding marginal density path.
  • Gaussian Flow Matching is essentially equivalent to the corresponding diffusion formulation when the Gaussian schedule satisfies the diffusion-SDE compatibility condition.Under a linear schedule, Gaussian Flow Matching can also be interpreted as diffusion trained to predict velocity.
  • Although conditional interpolations are straight, the induced marginal ODE trajectory is generally not affine in time.Changing endpoint couplings through Rectification or Reflow can make the marginal transport progressively closer to straight while preserving marginals and not increasing transport cost.
  • Repeated Reflow progressively refines the endpoint coupling and can produce transport paths that are easier to integrate with few numerical steps.The straightness measure need not improve monotonically at every iteration, despite the overall straightening tendency.

A Unified and Systematic Lens on Diffusion Models

Diffusion models share a density-level framework: forward processes define probability paths, while variational, score-based, and flow-based methods learn compatible reverse dynamics. Their formulations and oracle prediction targets are often algebraically equivalent, but schedules, weighting, finite model capacity, and numerical solvers can still affect practical behavior.

  • Unified principle: Variational, score-based, and flow-based diffusion perspectives converge on learning reverse dynamics for a forward path of marginal distributions.The shared framework connects discrete kernels, SDEs, and ODEs through density evolution.
  • Unified principle: The Fokker–Planck equation provides the common density-evolution constraint, even when deterministic and stochastic trajectories differ.Different dynamics can produce the same marginal path while following different sample trajectories.
  • Training objectives: Conditional KL, score, and velocity objectives differ from their original objectives only by model-parameter-independent constants, so they share minimizers within a model class.Conditioning makes the objectives tractable without changing their optimization targets.
  • Prediction equivalence: Noise, clean-data, score, and velocity predictions contain the same oracle information and can be converted into one another algebraically.Given the noisy observation and forward schedule, one oracle determines the others through the stated identities.
  • Affine equivalence: Affine schedules and parameterizations are mutually convertible through time reparameterization, scalar rescaling, and schedule-dependent weights.This includes VE, VP, FM, and RF affine interpolations, while velocity conversion may require an affine head involving x_t.
  • Practical distinctions: The canonical linear schedule with v-prediction yields a constant-magnitude target, but this theoretical property does not make it universally superior in practice.Performance also depends on architecture, normalization, loss weighting, samplers, discretization, guidance, regularization, data scaling, and training budget.

Guidance and Controllable Generation

The chapter presents guidance as a unified framework for controllable generation and preference alignment, covering classifier-based, classifier-free, and direct preference methods. These approaches trade conditioning strength, realism, diversity, training complexity, and sampling cost.

  • Guidance and controllable generation: Conditional scores decompose into an unconditional direction and a guidance direction that steers samples toward a condition while preserving realism.The framework distinguishes satisfying a condition from aligning outputs with human preferences.
  • Classifier guidance: Classifier guidance adjusts conditioning strength through ω and can reuse any pretrained unconditional diffusion model with an additional classifier.Increasing ω typically raises conditional fidelity, while relying less on the unconditional score typically reduces diversity.
  • Classifier guidance: Classifier guidance requires a classifier trained across noise levels, robust to severely corrupted inputs, and separately aligned with the diffusion model.These requirements create computational, robustness, and distribution-alignment limitations.
  • Classifier-free guidance: Classifier-free guidance provides a post-training trade-off between conditioning strength and sample diversity without a separately trained classifier or unconditional diffusion model.With exact scores, guidance weight 1 recovers the exact conditional score, while other strengths require distinguishing tilted densities from guided-dynamics marginals.
  • Preference alignment: Diffusion preference optimization remains computationally heavy because it requires O(T) per-step log-densities and backpropagation through stochastic chains for policy and reference paths.The text reports T ∼ 10^2–10^3 and notes that repeated sampling can exceed practical training budgets.
  • Preference alignment: DPO and Diffusion-DPO replace RLHF’s reward-model and reinforcement-learning stages with direct preference optimization from pairwise choices.Diffusion-DPO derives a tractable stepwise objective whose log-sigmoid loss has a DSM-style MSE-difference margin.
  • Sampling and denoising: Diffusion sampling after training is primarily a numerical integration problem, while denoising-based generation progressively refines predictions through repeated steps.InDI interprets diffusion modeling as supervised denoiser learning whose continuous-time limit yields the probability-flow ODE.

Coordinate Choice Is Part of the Numerical Method

DDIM’s finite-step behavior depends on the coordinates used before discretization, while its variational construction exploits freedom to change temporal couplings without changing one-time marginals.

  • Coordinate choice: Changing state and time coordinates can preserve the continuous ODE while altering its finite-step discretization.A suitable coordinate system can absorb analytically known dynamics before numerical approximation.
  • Coordinate choice: DDIM is explicit Euler in schedule-aligned coordinates, although it generally differs from plain Euler in (x, t).The noise-prediction form uses (y, ρ), while the clean-data form uses state normalization by σ_t and the corresponding α_t/σ_t time coordinate.
  • Variational perspective: The denoising objective fixes one-time perturbation marginals but does not uniquely determine couplings between noisy states at different times.Different transitions can share the same marginals while changing dependence across time.
  • Variational perspective: The marginal-consistent Gaussian family has one free noise coefficient c_t,s, with 0 ≤ c_t,s ≤ σ_t, after matching means and variances.The coefficient controls newly injected Gaussian noise while preserving the target marginal.
  • Variational perspective: The DDPM posterior is one marginal-consistent choice, whereas c_t,s = 0 yields the deterministic DDIM coupling and permits skipped intermediate times.The same trained denoiser can therefore be reused on a coarser sampling grid.

DDIM and CFM

DDIM and CFM arise from the same conditional Gaussian path but use different temporal views: DDIM takes finite-time steps, while CFM uses instantaneous conditional velocity. DEIS extends DDIM’s constant prediction with polynomial history.

  • DDIM and CFM: DDIM uses a finite-time map from the conditional Gaussian path, whereas CFM uses its instantaneous conditional velocity.Posterior averaging the conditional velocity gives the marginal PF-ODE velocity.
  • DDIM and CFM: A finite DDIM step is not generally the exact marginal PF-ODE flow because its model prediction remains fixed during the step.The exact marginal velocity changes continuously with the evolving state.
  • DEIS: DEIS replaces DDIM’s constant neural prediction with a higher-order polynomial extrapolation using evaluations from previous completed steps.Known schedule-dependent factors remain analytically inside the weighted integral.
  • DEIS: Under smoothness and bounded-step-ratio assumptions, full-history degree-n DEIS has formal order n + 1 with one-step local defect O(h^n+2).Warm-start steps increase from order 1 and reach full order n + 1 after n + 1 anchors; global guarantees additionally require stability and accurate starting values.
  • DEIS: AB-DEIS-0 is exactly the deterministic DDIM update and remains a first-order method.The index 0 denotes polynomial degree, not numerical order.
  • DPM-Solver connection: For a fixed diffusion schedule, DPM-Solver-1 and deterministic DDIM are identical first-order finite-step updates.Higher-order DPM-Solvers use the log-SNR coordinate to approximate neural-prediction variation.

DPM-Solver and Runge–Kutta Methods

DPM-Solver combines Runge–Kutta-style staging with analytic treatment of known diffusion dynamics, while DPM-Solver++ adapts the formulation for strongly guided sampling. Training-based distillation instead learns fast generators or flow maps directly.

  • DPM-Solver: DPM-Solver retains known diffusion-dependent weights analytically and approximates the learned prediction using within-step evaluations.Its connection to Runge–Kutta methods is structural rather than equality of finite-step formulas.
  • DPM-Solver++: DPM-Solver++ targets strong-guidance difficulties by using data prediction, enabling data-space corrections, and a multistep variant that reuses evaluations under a fixed NFE budget.Large guidance scales can amplify outputs and derivatives, narrowing high-order solvers’ convergence region.
  • DPM-Solver++: Without thresholding, data prediction is a pointwise algebraic re-expression of the same empirical PF-ODE, but finite-step approximation can still change.Thresholding is an additional model-output modification separate from numerical-order analysis.
  • DPM-Solver++: DPM-Solver++(2M) matches the degree-one weighted-polynomial update through second order, while 2S obtains second-order information from an intermediate stage.The two methods are generally different finite-step formulas.
  • Training-based acceleration: Distillation can align the student’s output distribution with the teacher’s or reproduce the teacher’s sampling path through flow-map learning.A learned map Ψ_T→0(x_T) with x_T ∼ p_prior enables one-step generation.
  • Training-based acceleration: Solver-based acceleration trades numerical integration cost against discretization error, whereas distillation trains students to replace many teacher steps with one or a few steps.The teacher may require hundreds of steps while the student can achieve comparable quality in only a few steps.

Question 10.3.1

Progressive distillation trains a coarse student step to match the endpoint of multiple fine teacher steps, then repeatedly halves the sampling grid. This compresses long diffusion trajectories into fewer transitions while preserving the time horizon.

  • DDIM Target: For DDIM, algebraically inverting the coarse update yields a closed-form pseudo-clean target for student training.The target is defined so a single step from s to t lands at the same endpoint as two teacher steps through u.
  • Iterative Halving: Each distillation round doubles the student time step and halves the number of transitions while retaining the same horizon [0, T].The procedure progresses from skip-2 to skip-4 and coarser grids until the desired inference-step count is reached.
  • Local Semigroup Matching: Progressive distillation matches one student jump to the teacher’s composition of adjacent fine-step transitions.The local semigroup relation replaces an intractable full rollout with short teacher fragments.
  • Generalized Solvers: The principle extends beyond DDIM when a coarse solver update can be inverted or when student and teacher endpoints can be directly matched.More complicated solvers may require backpropagation through the student solver instead of analytic inversion.
  • Generalized Solvers: Fewer time steps do not necessarily yield the same reduction in neural function evaluations when a coarse step requires multiple network evaluations.This limits the direct equivalence between fewer sampling transitions and lower computational cost.
  • Broader Context: Distillation can bridge iterative diffusion quality and one-step-generator speed, but it requires a pretrained teacher in the standard pipeline.The teacher is trained first and then distilled into a faster student; flow-map models pursue few-step generation without that dependency.

Question 11.1.1

The chapter develops standalone and distilled flow-map formulations that learn direct time-to-time mappings while connecting them to diffusion inference. These mappings support flexible sampling, but stochastic multi-step use can accumulate errors.

  • Standalone Training: Consistency Training requires no pretrained model and can be trained from scratch as a standalone generative model.Consistency Distillation instead uses a teacher for initialization and supervision.
  • Flow-Map Formulation: CTM learns finite transitions and infinitesimal diffusion dynamics within one parameterization.This combines long-jump flow-map behavior with direct access to local diffusion dynamics.
  • Flexible Inference: CTM learns Ψs→t for any s > t, enabling anytime-to-anytime transitions and flexible sampling strategies.Its parameterization also supports ODE solvers and exact likelihood computation.
  • γ-Sampling: When γ ≠ 0, iterative neural jumps introduce mismatches that accumulate toward time zero, degrading long multi-step sampling.The chapter reports degradation once the number of steps M ≥4.
  • γ-Sampling: With γ = 0, CTM empirically improves generation quality as sampling steps increase without introducing instability.Eliminating temporal overlap between transitions yields a tighter error bound.
  • Unified View: CTM and MF generalize diffusion inference to direct time-to-time mappings while retaining guidance, likelihood computation, and higher-order solvers.This compatibility follows from recovering instantaneous diffusion drift in the infinitesimal limit.

B.3 (Optional) Wasserstein Gradient Flows as Distribution-Level Training

Wasserstein gradient flow describes training as moving particles and their induced distribution toward the data distribution, then approximating that ideal motion with neural parameter updates. The framework connects distribution-level dynamics to likelihood and score-based training.

  • Distribution-Level Training: The distribution-level question is whether ideal particle motion toward pdata can guide neural-network parameter updates.The chapter explicitly separates the ideal distribution dynamics from their finite-dimensional neural realization.
  • Particle Dynamics: A training-time velocity field wτ moves generated particles, while the continuity equation describes the resulting evolution of the model distribution.Here τ denotes training time rather than diffusion or noising time.
  • Wasserstein Gradient Flow: Wasserstein gradient flow chooses particle velocity to decrease a discrepancy between the model and data distributions under 2-Wasserstein geometry.The first variation defines a local energy potential, whose negative spatial gradient gives the ideal particle velocity.
  • Neural Realization: Neural training projects the desired distribution-level particle motion onto movements realizable through shared finite-dimensional parameters.Generated particles are moved toward targets, and the network is updated to reproduce those moved particles.
  • Likelihood Training: Forward-KL minimization is equivalent to maximum likelihood when the data distribution is fixed.The chapter relates this distribution-level objective to likelihood-based models and diffusion likelihood or score-matching surrogates.
  • Score-Based Training: Reverse-KL-based methods such as DMD and VSD use the difference between target and model scores to update the generator.This score-difference gradient supplies a distribution-level descent direction on generated samples.

C.2. Change-of-Variable For Measures: Girsanov’s Theorem in Diffusion Models479

Girsanov’s theorem connects weighted score error to mismatch between exact and learned reverse-time path laws. This path-space discrepancy also bounds data-side KL divergence and expected negative log-likelihood.

  • Matching the full path law implies matching every time marginal, but identical marginals can arise from different trajectory couplings.The forward SDE and probability-flow ODE share marginal density paths while generally differing in sample trajectories and path laws.
  • The exact reverse-time SDE reproduces the forward diffusion’s time-reversed path law when initialized from pT and given the true score.This statement holds under standard regularity conditions.
  • Likelihood-weighted score matching minimizes reverse path-law KL up to the fixed terminal-distribution mismatch.The terminal term vanishes when pT = pprior; denoising and marginal score-matching objectives differ only by a parameter-independent constant, whereas other time weightings generally lack this interpretation.
  • The resulting path-space KL upper-bounds data-side KL divergence and expected negative log-likelihood up to a parameter-independent constant.This follows because endpoint projection and KL contraction cannot increase divergence under marginalization.

D.2.5 Theorem 4.2.1: Marginal Alignment via Fokker–Planck

Fokker–Planck equations establish that both probability-flow ODEs and reverse-time SDEs reproduce the intended marginal density path. Under uniqueness conditions, matching the PDE and initial distribution guarantees marginal alignment.

  • PF-ODE Marginal Densities: The probability-flow ODE transports an initial distribution through a flow map whose density satisfies the continuity equation.The flow map is defined by the velocity-field ODE, and its pushforward density evolves according to the corresponding PDE.
  • PF-ODE Marginal Densities: Under sufficient smoothness, equality of the continuity equation and initial condition implies that the ODE-induced density equals pt for every t.Uniqueness of the PDE solution yields ˜pt = pt on the full time interval.
  • PF-ODE Marginal Densities: Running the PF-ODE backward from pT preserves the marginals in reverse time under the same PDE uniqueness argument.The backward density and target density satisfy the same PDE with matching terminal initial condition.
  • Reverse-Time SDE Marginal Densities: The reverse-time SDE has the forward SDE’s marginal distributions in reverse order when initialized from pT.Its Fokker–Planck equation matches the reversed forward equation, with uniqueness completing the marginal-density argument.
  • Score Matching Optimum: The optimal squared-error score predictor is the conditional expectation given xt, which equals the marginal score ∇xt log pt(xt).This identifies the population minimizer of the score-matching objective under suitable regularity conditions.
  • Flow-Induced Density: Continuity-equation equivalence relies on regularity assumptions such as continuously differentiable, globally Lipschitz velocity fields and sufficiently smooth densities.These conditions support uniqueness through the method of characteristics.
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