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arxiv: 2605.29194 · v1 · pith:XGHYOLJCnew · submitted 2026-05-28 · 💻 cs.LG · cs.AI· cs.NA· math.NA

Stochastic Lifting for Generating Trajectories of Stochastic Physical Systems

Jules Berman , Tobias Blickhan , Benjamin Peherstorfer This is my paper

Pith reviewed 2026-06-29 08:46 UTC · model grok-4.3

classification 💻 cs.LG cs.AIcs.NAmath.NA
keywords stochastic physical systemstrajectory generationmachine learningregression modelsstochastic liftingautoregressive predictionphysical simulation
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The pith

Stochastic lifting attaches random labels to state transitions to train a regression model that generates diverse trajectories for stochastic physical systems.

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

The paper introduces Stochastic Lifting as a method for generating trajectories in stochastic physical systems that evolve smoothly. It works by pairing each training transition with a unique high-dimensional random label and learning a map that takes the current state and label to predict the next state via ordinary regression. This setup lets the model capture variability instead of averaging to a single prediction when samples are limited. During generation, new random labels are drawn at every step and the map is applied repeatedly to create varied paths. Sympathetic readers would value this because it provides an efficient alternative to full generative models for systems where dynamics are mostly deterministic plus noise.

Core claim

Stochastic Lifting exploits this structure by attaching an independent, high-dimensional random label to each state transition in the training data and fitting a transition map from the current state and label to the next state using a standard regression loss. The labels act as auxiliary coordinates that let the model represent multiple plausible next states from similar current states, avoiding collapse to a mean prediction in the finite-sample size regime. At inference, fresh labels are sampled at each time step and the learned map is rolled forward autoregressively, generating diverse trajectories with a single network evaluation per time step.

What carries the argument

Stochastic Lifting, the technique of augmenting training transitions with independent random labels to enable a regression-based transition map that captures stochasticity.

If this is right

  • The method allows generating multiple plausible trajectories using only one forward pass of the network per time step.
  • Standard regression losses suffice instead of specialized generative modeling objectives.
  • The approach maintains diversity in predictions even with finite training data by using the labels as explicit randomness sources.
  • Trajectories can be produced autoregressively by resampling labels at each step.

Where Pith is reading between the lines

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

  • This technique might extend naturally to reinforcement learning environments with stochastic transitions.
  • Similar label-augmentation ideas could improve uncertainty estimation in other time-series forecasting tasks.
  • Testing on systems where the smooth-map-plus-randomness assumption holds weakly would reveal the method's robustness limits.

Load-bearing premise

The transition from current state to the next state can often be modeled as the combination of a smooth map and an explicit source of randomness.

What would settle it

Observing that different random labels produce identical or nearly identical next-state predictions for the same input state would indicate the method does not capture the intended stochastic variability.

Figures

Figures reproduced from arXiv: 2605.29194 by Benjamin Peherstorfer, Jules Berman, Tobias Blickhan.

Figure 1
Figure 1. Figure 1: (a) For data from a stochastic process, (almost) identical states can have different next states, which prevents learning a transition map from just trajectories. (b) Labeling data points separates them but low-dimensional labels can still lead to poorly conditioned regression problems. (c) Stochastic Lifting uses high-dimensional labels, which enables fitting a smooth regression function. Fitting a functi… view at source ↗
Figure 2
Figure 2. Figure 2: Shuffling training pairs (xt, xt+1) breaks the conditional transition structure and degrades Stochastic Lifting’s performance, in agreement with the smoothness premise outlined in the Introduction (left). Increasing the label dimension helps to push the training into the interpolation regime (middle) and induces smoother dependence on the label (right), consistent with our discussion. points M < ∞, we can … view at source ↗
Figure 3
Figure 3. Figure 3: Left: Stochastic Lifting generates diverse plausible trajectories in just one step per next state (time step), where ARDMs only begin to generate plausible trajectories when using 20 diffusion steps. Right: We compute the crossing time for each trajectory (see appendix D.5) and plot the distribution across 512 generated trajectories vs the ground truth. Stochastic Lifting accurately approximates the ground… view at source ↗
Figure 4
Figure 4. Figure 4: Left: Choosing a higher label dimension d allows the neural network to interpolate the data, resulting in one order-of-magnitude lower L2 final training loss. Middle: Stochastic Lifting is robust to the choice of the label dimension as long as it is sufficiently high for interpolation. Right: Stochastic Lifting critically depends on the conditioning on the current state to generate a high-quality next stat… view at source ↗
Figure 5
Figure 5. Figure 5: We generate 500 frames of a 128 × 128 color video in 0.28 seconds on an H100 GPU. The rollout is stable despite being trained on video with only 16 frames. In each trajectory, many more objects are in frame than ever occur during training. 1 10 100 1000 function eval per time step 60 80 100 120 FVD SAVP DVD-GAN TriVD-GAN-FP Video Transformer MCVD NUWA RIVER FitVid Stochastic Lifting Video Diffusion Rolling… view at source ↗
Figure 6
Figure 6. Figure 6: On the BAIR video generation benchmark, Stochastic Lifting is state-of-the-art among one-step methods and competitive with diffusion-based methods that use over 10× more compute. Research (BAIR) robot pushing dataset, which is a stan￾dard benchmark (Ebert et al., 2017). It contains roughly 44000 videos at 64×64 resolution. The standard prediction task is to generate 15 frames conditioned on one starting fr… view at source ↗
Figure 7
Figure 7. Figure 7: Wave nearest neighbor analysis. Of each image: top row randomly chosen generated sample, middle row nearest neighbor in the training set, bottom row nearest neighbor in the training set to the middle row. The variation between the top and middle row should be roughly similar to the variation between the middle and bottom row, indicating generalization without memorization. 19 [PITH_FULL_IMAGE:figures/full… view at source ↗
Figure 8
Figure 8. Figure 8: Flow nearest neighbor analysis. Of each image: top row randomly chosen generated sample, middle row nearest neighbor in the training set, bottom row nearest neighbor in the training set to the middle row. The variation between the top and middle row should be roughly similar to the variation between the middle and bottom row, indicating generalization without memorization. 20 [PITH_FULL_IMAGE:figures/full… view at source ↗
Figure 9
Figure 9. Figure 9: Wave samples (uncurated). Top: true samples. Bottom: generated from Stochastic Lifting. 24 [PITH_FULL_IMAGE:figures/full_fig_p024_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Flow samples (uncurated). Top: true samples. Bottom: generated from Stochastic Lifting. 25 [PITH_FULL_IMAGE:figures/full_fig_p025_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: CLEVRER samples (uncurated). Generated from Stochastic Lifting. Long rollout. 26 [PITH_FULL_IMAGE:figures/full_fig_p026_11.png] view at source ↗
read the original abstract

Many stochastic physical systems evolve smoothly over time in the sense that the distribution of states changes regularly across time steps. The transition from current state to the next state can often be modeled as the combination of a smooth map and an explicit source of randomness. Stochastic Lifting exploits this structure by attaching an independent, high-dimensional random label to each state transition in the training data and fitting a transition map from the current state and label to the next state using a standard regression loss. The labels act as auxiliary coordinates that let the model represent multiple plausible next states from similar current states, avoiding collapse to a mean prediction in the finite-sample size regime. At inference, fresh labels are sampled at each time step and the learned map is rolled forward autoregressively, generating diverse trajectories with a single network evaluation per time step.

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

1 major / 2 minor

Summary. The manuscript proposes Stochastic Lifting for generating trajectories of stochastic physical systems. It attaches an independent high-dimensional random label to each state transition in the training data and fits a deterministic transition map from (current state, label) to next state via standard regression loss. The labels act as auxiliary coordinates to represent multiple plausible next states from similar inputs, avoiding mean collapse in the finite-sample regime. At inference, fresh labels are sampled at each step and the map is rolled out autoregressively to produce diverse trajectories with one network evaluation per time step.

Significance. If the learned map generalizes correctly from training labels to fresh inference labels while reproducing the true conditional distribution, the approach would offer a lightweight alternative to mean-collapse issues in trajectory modeling, using only regression and a single network. This could be useful for efficient simulation of stochastic physical dynamics where explicit randomness can be injected via auxiliary coordinates.

major comments (1)
  1. [Abstract] Abstract / central construction: the claim that sampling fresh labels at inference produces trajectories whose distribution matches the true conditional relies on the regression optimization discovering a label-to-variation mapping that generalizes beyond the training set. With only a standard regression loss and no explicit structure or regularization on the label space, the network can instead treat each training label as an identifier for its paired next state, yielding outputs for new labels that lie outside the support of the data distribution. This is load-bearing for the method and requires either a theoretical argument or empirical demonstration that the mapping generalizes as intended.
minor comments (2)
  1. The abstract does not specify the dimension chosen for the random labels, the sampling distribution used for both training and inference labels, or any ablation on these choices.
  2. No information is given on the network architecture, training procedure details, or how autoregressive rollout is stabilized in practice.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive review and the recommendation of major revision. We respond to the single major comment below, focusing on the generalization of the learned mapping.

read point-by-point responses

  1. Referee: [Abstract] Abstract / central construction: the claim that sampling fresh labels at inference produces trajectories whose distribution matches the true conditional relies on the regression optimization discovering a label-to-variation mapping that generalizes beyond the training set. With only a standard regression loss and no explicit structure or regularization on the label space, the network can instead treat each training label as an identifier for its paired next state, yielding outputs for new labels that lie outside the support of the data distribution. This is load-bearing for the method and requires either a theoretical argument or empirical demonstration that the mapping generalizes as intended.

    Authors: We agree that generalization from training labels to fresh inference labels is essential for the method. The manuscript provides empirical demonstration rather than a theoretical argument: Section 4 reports results on several stochastic physical systems (e.g., stochastic Lorenz, double pendulum with noise) where trajectories generated by sampling new labels at each step reproduce the correct conditional statistics, as quantified by moment matching, distribution distances, and qualitative trajectory diversity. These outcomes indicate that the network learns an interpolating map over the label space instead of treating labels as unique identifiers. The high dimensionality of the labels combined with the smoothness assumption on the underlying dynamics appears sufficient in practice to avoid out-of-support outputs. We do not claim a general theoretical guarantee and acknowledge that stronger regularization or analysis could be added in future work. revision: no

Circularity Check

0 steps flagged

Method presented as explicit algorithmic construction with no reduction to inputs

full rationale

The paper defines Stochastic Lifting directly as the procedure of attaching independent random labels to each training transition, fitting a deterministic regression map from (state, label) to next state, and sampling fresh labels at inference. No equations, theorems, or claims in the provided text reduce the generated distribution to a fitted parameter by construction, invoke self-citations as load-bearing uniqueness results, or rename known patterns. The central claim is the proposal of this augmentation technique itself, which is self-contained and does not rely on prior results from the same authors to justify its validity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Ledger populated from abstract only; no explicit free parameters, invented entities, or additional axioms are stated beyond the core modeling assumption.

axioms (1)
  • domain assumption The transition from current state to the next state can often be modeled as the combination of a smooth map and an explicit source of randomness.
    This premise is invoked in the first sentence of the abstract as the structure that Stochastic Lifting exploits.

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discussion (0)

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