Dynamic Diffusion Schrödinger Bridge in Astrophysical Observational Inversions
Authors: Ye Zhu
Duo Xu
Zhiwei Deng
Jonathan Tan
Olga Russakovsky
Venue: NeurIPS 2025 poster ()
Keywords: Diffusion Schrödinger bridge
Astrophysics
ML4Science
Primary area: machine_learning_for_sciences
TL;DR: We introduce Astro-DSB tailored for astrophysical observational inverse predictions within Giant Molecular Clouds (GMCs) for star formation.

Abstract: We study Diffusion Schrödinger Bridge (DSB) models in the context of dynamical astrophysical systems, specifically tackling observational inverse prediction tasks within Giant Molecular Clouds (GMCs) for star formation. We introduce the Astro-DSB model, a variant of DSB with the pairwise domain assumption tailored for astrophysical dynamics. By investigating its learning process and prediction performance in both physically simulated data and in real observations (the Taurus B213 data), we present two main takeaways. First, from the astrophysical perspective, our proposed paired DSB method improves interpretability, learning efficiency, and prediction performance over conventional astrostatistical and other machine learning methods. Second, from the generative modeling perspective, probabilistic generative modeling reveals improvements over discriminative pixel-to-pixel modeling in Out-Of-Distribution (OOD) testing cases of physical simulations with unseen initial conditions and different dominant physical processes. Our study expands research into diffusion models beyond the traditional visual synthesis application and provides evidence of the models' learning abilities beyond pure data statistics, paving a path for future physics-aware generative models which can align dynamics between machine learning and real (astro)physical systems.

Dynamic Diffusion Schrödinger Bridge in Astrophysical Observational Inversions
Ye Zhu Duo Xu Zhiwei Deng', Jonathan C. Tan Olga Russakovsky
Department of Computer Science, Princeton Univeristy, Princeton NJ, USA LIIX, École Polytechnique, IP Paris, Palaiseau, France Canadian Institute for Theoretical Astrophysics (CTTA), University of Toronto, Toronto, Canada Google DeepMind, Mountain View, CA, USA
Department of Astronomy, University of Virginia, Charlottesville VA, USA
6 Department of Space, Earth & Environment, Chalmers University of Technology. Gothenburg, Sweden ye. ye.zzu@@polytaccmiqque edu, xuduoucite ta. www.unto.ooo... ca, zhiweidengdrooglle com, wwwtteeviiiiniee edu, olgarus@princeton..ee
Abstract
We study Diffusion Schrödinger Bridge (DSB) models in the context of dynamical astrophysical systems, specifically tackling observational inverse prediction tasks within Giant Molecular Clouds (GMCs) for star formation. We introduce the Astro- DSB model, variant DSB with the pairwise domain assumption tailored for astrophysical dynamics. By investigating its learning process and prediction perfor- mance in both physically simulated data and real observations (the Taurus B213 data), present two main takeaways. First, from the astrophysical perspective, our proposed paired DSB method improves interpretability, learning efficiency, and prediction performance over conventional astrostatistical and other machine learning methods. Second, from the generative modeling perspective, probabilis- tic generative modeling reveals improvements over discriminative pixel-to-pixel modeling in Out-Of-Distie Distribbtbtoon (OOD) testing cases of physical simulations with unseen initial conditions and different dominant physical processes. Our study expands research into diffusion models beyond the traditional visual synthesis application and provides evidence of the models' learning abilities beyond pure data statistics, paving path for future physics-awar generative models which can align dynamics between machine learning and real (astro)physical systems.
1 Introduction
Diffusion models (DMs) (Sohl-Dickstein et al. 20155 Ho et al. 2020 Song et al. 2020). sometimes also known as score-based generation models (SGMs), have become a popular probabilistic modeling approach in machine learning for many data generation applications (Dhariwal and Nichol 2021 Ho
al. 2022 Rombach al. 2022). recent line of work improves vanilla design by reformulating the mapping trajectory learning in the context of the Schrödinger bridge problem (Schrödinger 1932)
restrictions on the learned trajectory, leading Diffusion Schrödinger Bridge models (DSB) (Wang et al.) [221]] De Bortoli et al. 2021 Shi et al. (023). These DSB models are theoretically grounded and have proven effective in vision-related applications such as image generation and translation (Liu et al. [2023] Gushchin al.) 2024). However, their modeling capabilities and limits could extend beyond vision, offering a new opportunity to test these generative
which allows for imposing domain-spennt
WWrk mainly completed when YZ was postdoc at Princeton University. "ode available https://ittub..co/LLWWCe
http://hhwbbb.oooolo//////ttttoo YeZhh//AstroDDB
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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ID Prediction
OOD Prediction
Observation
Our Predicttion
Graud Truth
Obsersation
Prediction
Groun Toush
Doustt
MagB
Figure 1: We introduce Astro-DSB for astrophysical inverse predictions from observables within Giant Molecular Clouds (GMCs). We visualize the observables, predicted results, and ground truth physical distributions the density (top) and magnetic field (bottom) for in-distribution (ID) and distii--iittiittinnn (OOD) testing cases, with the OOD defined via changes in initial physical conditions and dominant dynamical processes in the simulations.
formulations in the context of real-world physical systems, where data are governed by complex dynamics and observational constraints rather than pixel statistics.
In this work, we study a real-world physical problem, focusing on key class of inverse prediction tasks in astrophysics, particularly within the context of star formation (McKee and Ostriker, 2007)] Larson 2003 Madau and Dickinson 2014). Specifically, we aim to better understand and predict the physical states within Giant Molecular Clouds (GMCs) (Shu et al., 1987). which is a major component of the interstellar medium (ISM) (Spitzer 2008) that directly regulates star formation rates. The goal of our task is to infer true physical distributions, such as density or magnetic fields, from partial observables, as illustrated in Fig.
As critical and active research area in astrophysics, multiple methods have been developed over the past decades to tackle this challenge. Given that observational inverse prediction problems are inherently intractable, early approaches often relied on astrostatistical methods such as power-law fitting (Maschberger and Kroupa| 2009), which combine theoretical simplifications with observational statistical priors (detailed in Sec. 2.2 and Sec. 4.2p. However, these methods typically suffer from large prediction errors and limited generalization on out-of-distribution (OOD) testing cases under shifted physical initial conditions, due to often oversimplified physical assumptions and restricted capacity for modeling complex non-lineasities.
Given the increasing capability of neural networks in modeling large-scale data with highly compli- cated nonlincarities. the astrophysical community has recently started to explore machine learning approaches (Yang et al. 2024 Xu et al. (2200a) to overcome previous limitations, Among multiple ML architectures, conditional diffusion generative models (DGMs) (Sohl-Dickstein al. [0155] Ho et al. 20200 Song et al., (020)) stand out as recent state-of-the-ee art (SOTA) frameworks with large improvements in prediction accuracy over conventional methods for several critical physical quanti- ties, such as GMCs density distribution (Xu et al. 2023b) and magnetic field strengths (Xu et al. 2025). These methods typically take observables as input conditions and output the predicted physical distribution maps through the distribution mapping ability of DGMs trained on simulation data. However, despite the enhanced prediction accuracy, such conditional diffusion models share a few constraints, including physics-ML misalignment due to the Gaussian prior from the original DM formulation (also detailed in Sec. 3.2p, as well as the known expensive computational cost for learning DGMs.
To this end, we introduce Astro-DSB, a variant of Diffusion Schrödinger Bridge model tailored to astrophysics with better physics-ML state aligmment and improved learning efficiency. Specifically, compared to existing DGMs-based methods that learn mapping between the Gaussian prior and the true physical distribution with the observable the condition, Astro-DSB removes the reliance on the Gaussian prior from the DGMs formulation and learn the direct mapping between observables and ground- touuh physical states. This design also helps to accelerate the learning efficiency by 75%

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compared to the previous DGM-based SOTA. More concretely. Astro-DSB leverages the intrinsic deterministic mapping property between the observation and ground truth physical state and adopts a paired assumption between two boundary d Within this high-level Schrödinger bridge MI. framework, we further integrate the observables as an enhanced condition to facilitate the bridge learning process. Intuitively, this can be mathematically considered as learning conditional Doob's 's h-transform, thus differentiating from the vanilla paired DSB models in computer vision tasks (Liu et al. 2023) and scientific applications such as protein designs (Didi et al., (023).
Our comprehensive experiments on both molecular density and magnetic field tasks validate the effectiveness of our proposed method. In addition to the comparable or superior SOTA performance in ID testing cases, we also examine the performance under distribution shifts using both physically simulated data and real astrophysical observations from Taurus B213. In contrast to conventional vision tasks, where OOD is often heuristically defined, astrophysical systems provide scientifically meaningful definitions of distributional shifts through variations in initial conditions and dominant physical processes (Wang and Abel 2009, Wu et al.] 2013). These configurations reflect realistic physical diversity, making them especially suitable for testing distribution-level generalization. Through this evaluation, we uncover the practical boundaries of probabilistic modeling and its advantages in scientific domains such as astrophysics.
The rest of the paper is organized as follows: we present related works in Sec. 2 Sec. detaails the core technical designs; Sec. 4 #eeplains our controlled experiments and comprehensive analysis comparing against conventional astrophysical and existing ML approaches; and we conclude by discussing the limitations and future directions in Sec. [5] To summarize our main contributions, our work offers two key takeaway messages: for scientific applications such as astrophysical inverse prediction, the proposed Astro-DSB framework offers an efficient and accurate alternative to conventional modeling approaches; for the broader generative modeling community, our study highlights the advantage of distribution-level learning in enabling meaningful generalization under physically grounded distribution shifts, extending the application scenarios from simulation data to real-world observations from Taurus B213 (Li and Goldsmith 2012 Palmeirim et al. 2013).
2 Related work
2.1
Schrödinger bridge problem and generative modeling
The machine learning method we leverage this work is closely related to one line of active research efforts in the Schrödinger bridge problem (SB) with application to generative models. Specifically, given the similar underlying problem of learning the mapping between two arbitrary probabilistic distributions, prior works (Wang et al. [021] De Bortoli et al. 202T Shi et al. 2023
2022, Somnath et al., 2023 Liu et al.) 2023) have explored improving generative modeling by formulating the boundary distribution matching problem within the Optimal Transport (OT) map of SB (Cuturi. 2013). Specifically, [Wang et al. (021) proposes to learn a generative model via entropy interpolation with a Schrödinger bridge in two-stage manner. While this work relies on static SB problem, later works (De Bortoli et al. 2202 Shi et al., 2023 Liu et al. [023] Gushchin et al. 2024 Ksenofontov and Korotin 2025) explicitly propose to compute the dynamic SB with the additional time moments / E [0, IJ via the score-based diffusion models to learn the desired optimal path, under different specific restrictions (e.g., continuous v.s. discrete data space and unpaird v.s. paired domain translation). These prior works largely target static domains in practical applications, lcaving open the question of how DSB performs settings where the dynamic trajectory is physically meaningful, which is question we directly investigate in this work.
2.2 Inverse astrostatistical predictions for star formation
The real-world physical problem we address this paper focuses on a key class of inverse prediction tasks in astrostatisties, particularly within the context of star formation (McKee and Ostriker [007]] Larson 2003 Madau and Dickinson, 2014). We focus on the dynamic physical processes within Giant Molecular Clouds (GMCs) (Shu et al. 1987). major component of the interstellar medium
'Terminology note: The term "paired DSB" is used here in the ML sense, which refers to the sample-leve conditioning between observables and physical states, rather than in the classical unpaired Schrödinger Bridge formulation based probabilistic coupling.
3

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(ISM) (pitzer Jr 2008) that directly regulates star formation rates. The goal of our task is to infer true physical distributions, such as density or magnetic fields, from partial observables. In contrast to image generation or translation tasks where input-output mappings are static and visually structured, our problem setup reflects dynamic and physically driven distributional transitions.
Traditionally in astronomy, such observational inverse prediction problems have been commonly approached via astrostatistical methods, which often combine theoretical simplifications with statis- tical priors (also in Sec. 4 For instance, molecular density is commonly estimated using two- or toreecemmpnnet power-lat law conversion derived under idealized physical assumptions about emission properties and cloud structure (Bisbas et al.] 2021 2023). While effective within limited regimes, these methods tend lack generalization ability when applied to systems governed by varying physical conditions and dynamics, making their performance unreliable in unseen testing cases.
Grounding DSB modeling within this context thus enables more meaningful evaluation of proba- bilistic models beyond synthetic benchmarks QLiu et al. 2023] Gushchin et al., 2024 Ksenofontov and Korotin 2025 Su al. 2023). Notably, we leverage a key strength of the astrophysical setting: the ability to define Out-of- croroff-Ditribttion (OOD) conditions in principled and physically grounded way through simulations with varying initial conditions and dominant physical processes. In addition to standard evaluation on In-Domain (ID) cases, our experiments explicitly assess OOD generalization, offering new insights into the robustness of probabilistic modeling in scientific prediction tasks.
3 Diffusion Schrödinger bridge for astrophysical observational inversion
3.1 Problem statement and preliminary in astronomy and ML
GMC's observational prediction. Studying the precise physical distributions within Giant Molec- ular Clouds (GMCs) is challenging yet essential for understanding the star formation process in astronomy. This challenge mainly arises from the highly complex and nonlinear physical dynamics within the forward evolution and the observational constraints.
Given a physical initial condition Xo. we assume that the GMCs evolve under a set of governing physical laws as modeled in previous astrophysical studies (Wu et al. 2015 2020 Hsu et al. (023)) This results in an equilibrium physical state X1, which we denote as X1 H( where is highly complex, nonlinear function incorporating self- eeff-gravity magnetic fields, radiative heating, and cooling processes. While the forward process H. is typically well-defined in controlled simulations, direct access to x1 is rarely available in practice. Instead, we often only observe limited projection y of the true physical state. Therefore, our objective is to infer the full distribution X from partial observable y, featuring class of observational inverse prediction problems that lie at the core of many tasks in astrostatistical modeling. We further assume an observational model in the form of:
y = F(x,) + t F(H(xo)) + e,
where F represents the observational process (e.g., line-of-sight integration, instrumental response) and & denotes corresponding observational noise in the measurement. Given this formulation, our core task becomes a conditional probabilistic inference problem, where the goal is estimate the posterior distribution yy).
Preliminary on dynamic DSB. Consider two distributions po, and a reference path measure Q E P(C), the dynamic SB problem seeks find a path measure pSB e P(C) such that:
PSB argming (KL() Fo po. PI Pi}.
with C = C([0, RR) defined as the space of continuous functions from time space [o. 1] to IR SB can be formalized as entropy-regularte OT problem describing the following forward and backward SDEs:
dx, = [f t) dx, (x(, p(t)) ())dt + g({)dw.
t)]dt + g(t)dw,
Here, f(,,t) is the linear drift, gt) controls the diffusion intensity over time, and dw represents the standard Wiener processes for modeling stochastic noise. The wave functions v and v are

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time-varying energy potentials that solve the following coupled PDEs:
(v 1) Vxx(x f)x,xx) 2 2 ( t),
Ot
a (x, t) V (ύ(x, (x, + (g( (x.
Ot
with (x, (x, = px(x) and (x, (x, px) Recent works have shown that score- based diffusion models provide a practical numerical solver for the DSB problem by parameterizing the time-dependent scores associated with forward and backward SDEs (De Bortoli et al.) 2021 Shi et al. 2023: Liu et al.
2023). Specifically, instead of explicitly solving the Schrödinger system via
wave potentials V/ and V SGMs learn the forward and reverse score functions Vlog px(x) through denoising score matching, simulating the optimal stochastic process by integrating the learned SDEs, This formulation enables the DSB problem be learned from data via stochastic sampling and bypasses the need for explicit access to density functions pr(x), which are often intractable in real-world scientific applications. In our work, build on this framework and introduce pairwise- constrained variant of the DSB model tailored to astrophysical observational inversion tasks, as described in the following section.
3.2 Diffusion Schrödinger bridge for astrophysical observational prediction (Astro-DSB)
Core technical motivation and relation to prior works. Our core technical motivation seeks to better align the states X, score-based diffusion models with the dynamic evolution processes governing GMCs, which leads to two key differences from existing literature.
Within this field of observational prediction in GMCs, existing sate-of-theear (SOTA) MI--based methods (Xu et al., 2223a b. (2025) leverage conditional diffusion denoising probabilistic models (DDPMs), framing the observational data y as the condition. While achieving superior performance compared to traditional astrostatistical methods, conditional DDPMs introduce unnecessary Gaussian prior assumptions, which are not naturally aligned with the physics of GMCs. Our proposed Schrödinger bridge framework removes this assumption by directly modeling the transition between observable y and the true physical distribution X1, resulting in iistribution-lever probabílistic mapping withoul reliance artificial prior structures, thus offering better ML interpretability compred to conditional DDPMs.
From the physics-aligned modeling perspective, Li et al. 2025) recently propose another variant of diffusion bridge with a two-stage learning process (data-driven pretraining and physics-aligned finetuning) for physical field reconstruction and super-resolution. In contrast, our approach adopts single-stage learning framework to learn the transition path directly without pretraining, and sys- tematically evaluates both in-distribution performance and OOD generalization to assess robustness, further extending the prediction to real observational data.
M0: Pairwise matching in marginal distributions. Similar to many other physical tasks (Li cet al. 2025), our observational prediction problem exhibits a pairwise connection between marginal distributions Po and pl established by y and x1, respectively. Based on this, we follow the prior work FSB (Liu et al. 2023) for a special case of the SB problem. Specifically, I'SB presents a tractable class of SB that eliminates the couplings of the wave functions from Eq. 3 and propose analytical form for the implementation of the posterior distribution:
o²
0.0² of X1, o; + of
p(x (x) = (Xi;
I),
(5)
o7 +
o
0
where of- : 1/ d, and o² = |:' 3, denote accumulated variances. This forms the base design
of our proposed Astro-DSB. However, we observe that this baseline framework alone is insufficient for accurately predicting true physical distributions from constrained observational inputs.
MI: Noise perturbation alignment between ML and observables. Compared to conventional image translation (Liu et al.) 2023) and other physical tasks such as field reconstruction (Li et al. 2025) that take a clean data input y. we note that explicitly modeling the noise perturbation term in observables y in Eq. offerrs simple yet critical alignment to physical assumptions in observational
5

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GT X1
Astro-DSB training
Clean obs. y
Fe. = (x. y-
at
Noisy obs.
Inter. state
U-Net
Fo.
Prediction
60. 0,t)
t
X
C 2
Aggregated Scalable Sampling
y ODs
Ei-1
yobs E1
E1+1
X
Figure 2: Illustration pipeline of our proposed Astro-DSB method. In the training stage, we propose learn the DSB model under the pairwise matching assumption with observational noises alignment and enhancement observables y. In inference, we adopt an aggregated scalable sampling strategy infer physical distributions from larger observables, such as Taurus B213.
prediction scenarios described in Eq. In this work, assume that the noise introduced from the observation process follows a weighted Gaussian, denoted as ~ (V f). Following statistical conventions in astrophysical studies, we ensures that our Astro-DSB framework respects the uncertainty structure of constrained observables, refining the targeted conditional posterior distribution be p(xi -
0.1 throughout the experiments. This modification
E).
M2: Observable enhancement within DSB model. Compared to the vanilla FSB framework, further propose enhance the learning process by incorporating the clean observable input as an additional intermediate condition at each time step t. This modification aims to facilitate the learning of the score function 68,r, implemented through U-Net architecture (Ronneberger al.) 2015), by consistently "reminding" the model of the observational constraint throughout the trajectory. Empirically, we find this design critical for achieving robust prediction performance within the DSB framework and helps simultaneously obtain lower loss values with faster convergence speed.
Patch-based training and aggregated scalable sampling. Following the methodological designs described above, we train the proposed Astm-DSBB using the reparameterized objective function:
Fe. = [Ea Xt, y, t; y
(6)
where X; takes the analytical form from Eq. 5 Given that astrophysical data often present large spatial resolutions, adopt patch-based training strategy that is widely used in both vision and scientific applications (Li et al. 2025 Yang et al. 2019), with cropped patches from the original simulation data. Specifically, Xı and y cropped into smaller patches of consistent size facilitate efficient learning while preserving local physical structures.
For inference, in contrast to the padding-based techniques used in prior works (Li et al. 2025), we design an aggregated sampling strategy to restore higher-resolution prediction results. Given large testing observable Yobe (c.g., real-world survey data), partition Yobe into overlapping patches

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Yobe that match the training resolution. We then independently run inference on each patch to obtain corresponding predictions x and aggregate across all patches to reconstruct the final predicted distribution X1. Fig. Dillustrates the described pipeline, with detailed algorithms in the appendices.
Experiments
4.1 Astrophysical data
) our training data are drawn from synthetic magnetohyddrodynamis (MHD) simulations. Specifically, we follow the simulation setup from Wu et al. (2020) and Hsu et al. (223). which are conducted using the DISCLL method and HLLD Riemann solver in the adaptive mesh refinement (AMR) code Enzo (Dedner et al. 2002 Wang and Abel 2009 Bryan et al. 2014). These simulations model key physical processes including self-gravity, magnetic fields, radiative heating, and cooling, based on photodissociation model (Wu et al. 2015). We consider the molecular density (p. measured in the number of H nuclei fe/ccm= ') and magnetic field strength (B, measured in juG as the primary quantities for prediction in this work. For molecular density, we collected data from 7179 simulation trials, split into 5707 training and 1472 testing cases following an 8:2 ratio. For magnetic field strength, obtained 19100 training and 6370 testing cases. All data are pre-processed by logarithmic transformation followed by normalization to [0.1], J. and resized to 128 x 128 patches.
Magseso simulations. Similar to previous work (Xu et al. 2023b 2025
Additionally, we generate 1680 independent tests with different dominate physical process assump- tions as the OOD testing cases to evaluate generalization. Specifically, unlike the ID MHD simulations code ENZO (Bryan al. 2014) initialized with colliding and non-colliding molecular clouds, OOD data are produced using a different MHD code, ORION2 (Li et al. 2021), which excludes gravity but includes driven turbulence, and employs a different set of initial conditions. While the fundamental MHD equations are similar between two simulation codes, the physical processes and initial setups differ significantly, thus clearly distinguishing the ID and OOD cases under physically meaning shifts.
Synthetic and real observables. After generating the simulation data, we define the observables y following conventions in astrophysical studies. For molecular density, we adopt the projected mass surface density maps (also known as the "column density". measured in NH/ cmm -3) as the observables y,. For the magnetic field, the synthetic observable yg consists of the column density, the orientation angles of the dust continuum polarization vector, and line-of-sight (LOS) nonthermal velocity dispersion. In addition to these synthetic observables, also evaluate the proposed Astro- DSB on real observational data from the Taurus B213 region, using column density maps provided by [Palmeirim et al. (2013). B213 is one of closest star-forming filamentary structures located in the Taurus molecular cloud (Li and Goldsmith 2012). Unlike the synthetic data where ground-trute distributions are available, the Taurus B213 observation 9213 lacks a precise reference for physical states, and is instead constrained by indirect measurements from complementary observations. This real-data evaluation demonstrates the applicability of Astro-DSB in practical scenarios where ground truth is unavailable, and highlights its potential for scientific discovery in observational settings.
4.2 Comparing methods and evaluations
Astrostatistical methods. The conventional methods involve astrostatistical techniques to infer physical distributions with simplified physical assumptions. For molecular density prediction, existing popular techniques include statistical conversions such as 2-component (2PLF) and 3-component plwer-law fitting (3PLF). For magnetic field, Davis Chandrasckhar-Feem (DCF) method (Davis) 1951 Chandrasekhar and Fermil 1953; Beck 2016) has been the classic method for estimating the magnetic field strength, which relies on the assumption of an equipartition between the magnetic field energy and the turbulent kinetic energy of the gas.
Machine learning methods. Recent studies have also adopted machine learning techniques, which can be broadly categorized into discriminative and generative approaches. Discriminative models such as convolutional neural networks (CNNs) trained with reconstruction losses have been widely adopted (Xu et al. 2020alb) b). From the generative modeling perspective, conditional Denoising Diffu- sion Probabilistic Models (DDPMs) (Ho et al. 2020) have emerged as a recent popular alternative to tackle those observational prediction problems.
7

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Table 1: Quantitative evaluation of density prediction results reported on the original data scale (100^"). The signs indicate the direction of prediction bias: negative denotes overprediction. and positive indicates underprediction. We bold the best results and underline the second-best i and a are the primary criteria for assessment, where values closer to zero indicate superior performance.
plug//m ")
ID
OOD
Methods
#- P-Papio (L))
--Poo (LD
ME
H. P-Paofo ((D)
0 O-Paole (4J)
ME
2PLF
-2.50-76-4
7.11:221-3
-715.95
-5.774336-4
7.21:11B-3
1680.82.10.7
3PLF
-0.77+55-4
4.44+-2F-3
-715.78=11.8
-3.23+22-4
4.21+7E-4
-6880.60=00.1
U-NET
10.25+18-4
1.29+-4F-4
118.24+0..4
6.01++6F:4
7.08++18-3
1313.60+0.66
cDDPMs
-0.55:16-5
0.61:16B-4
4.61:1..05
2.88:15E.4
5.33-17B-
1050.511:..61
Astro-DSB (Ours)
-0.02+1E-4
0.71+33F44
32.09+00.65
0,51+2F-4
2.32+-3E-3
-5.04+1.32
In our experiments, we compare with both conventional and ML-based methods. Specifically for ML, in addition to the state-of-the-art conditional DDPMs, we also re-implement a U-Net (Ronneberger al. 2013) from CASI-2D Xu et al. 2020a) as a pixel-to-pixel approach for comparison. Notably, while recent works suggest that generative models offer improved sample quality in image synthesis, their advantage scientific prediction, especially under physically grounded distribution shift (OOD), remains unclear. Our study explicitly tests this hypothesis by comparing probabilistic and pixel-to- pixel modeling under controlled simulation setups with similar model scales.
Evaluations. Our evaluation adheres to established protocols in astrophysical studies, emphasizing the distributional patterns of relative error between predictions and ground truth values. Specifically, we assess the normalized relative error (c.g.. Ap/p) p), where an ideal prediction should produce a distribution centered around zero Although this distribution may not strictly follow Gaussian form, a symmetric shape with narrow spread around zero indicates accurate and well- calibrated predictions. In contrast, distributions with large bias or heavy tails reflect systematic under- or over- esttimmttni To quantitatively capture these characteristics, we report the mean (gt pred) and standard deviation a of the relative error distribution as primary metries. These values provide insights into the central tendency and variability of model performance across different test cases.
In addition to relative error metrics, we also report the Mean Error (ME) between predicted and ground truth values. However, it is important to note that ME interpretation may be affected by the wide dynamic range of physical quantities in astrophysical studies. Specifically, even a method with a lower ME value does not necessarily indicate superior performance, as the absolute error can be dominated by regions of large values. For example, small relative error on value of 10? can contribute disproportionately to the ME compared to larger relative error on a value of 10. This imbalance highlights the importance of relative eтror patterns in accurately assessing performance.
Implementation details. To ensure a fair comparison among the tested MI. methods, we fix the U-Net backbone architecture across the discriminative U-Net, conditional DDPMs, and our proposed Astro-DSB, resulting in approximately 81M learnable parameters for each model. For the U-Net baseline, we train the model for 20 epochs using Mean Square Error (MSE) loss. For conditional DDPMs, we follow the setup from prior works (Xu et al. 2023a 2025) and adopt a 1000-step schedule with the standard variational lower bound loss (Ho et al.) 2020). Our proposed Astro-DSB is trained with the same time discretization (1000 steps) using the objective defined in Eq. [5] All models are trained with batch size of 16 using the AdamW optimizer with a learning rate of 5e-5, running on an AWS cluster with 4 NVIDIA T4 GPUs Notably. while conditional DDPMs require around 400 epochs converge, Astro-Dee DSB converges only around 100 epochs under the same setup, reducing overall training time by approximately 75%. This aligns with our expectations and observations in prior vision DSB works, as the target distribution P1 now lies closer to the physically meaningful observation distribution Po. compared to the case where Po is a zero-mean Gaussian in DDPMs. During inference, the runtime per testing case ranges from approximately 6 to 30 seconds, depending on whether the step skipping (Song et al.) 2021) is applied. Notably, we observe no significant difference in prediction performance across these sampler settings in our work.
4.3 Experimental results and analysis
Less biased prediction results for ID and OOD testing. We present the quantitative evaluation results for density and magnetic field prediction in Tab. and Tab. [ respectively. Our proposed DSB consistently achieves the best or second-best results across both ID and OOD settings,
8

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6OUPM-D
UNet-ID
AstroSBBIL
<DUPM-OOD
UNe-OOD
AstoLiB OOD
Prodiction results from di fforret MT. methork for ID and OOD acenarios, messured lng (ogr
cDDM
UNa
Astro DSB
diinn
10
8
-10
PDF comparison from setroctatisctall methods.
Extmple of residual berween OOD prediction and (nommalizad).
Figure 3: Multiple qualitative visualizations for relative error comparisons. (a) 2D histogram showing the distribution of ground truth (x-axis) versus predictions (y-axis). An accurate and unbiased prediction lies along the 1-to-1 diagonal line. (b) Relative error between the ground truth and predictions. (c) Residual maps showing the difference between model predictions and the ground truth across different methods. Results from Astro-DSB present more balanced and unbiased prediction error patterns compared to other ML methods, particularly for OOD testing cases. Best viewed in color and with zoom-in.
Table Quantitative evaluations of magnetic field prediction results reported on the original data scale (3E!~3). Bold values indicate the best results, underlined values indicate the second-best. Astro-DSB achieves competitive SOTA with only 25% of the computational cost of cDDPMs.
8 (GC) Methods
ID
OOD
R-P48/B (L)
0
P-P88/8 (1)
ME
[ Pa8/8 (1))
0-
-P88/8 (U))
ME
DCF
11.0018-4
21.78144-4
507.7410.03
5.19=8E-4
9.6413E-3
137.76:10.03
PLF
-0.11155-55
1.-4
-1.04:116-3
1.03-196-5
1.47-1E-4
17.14-12E-3
U-NET
-0,06112E-5
5.38-5B-5
154.7610008
-0.32-6B-5
0.7417E-5
146.38:10.02
cDDPMs
0.07-22-55
0.38=6E-5
123.99-0.04
0.12-7E-5
0.72+22-4
150.00:10.04
Astro-DBB (Ours)
0.15:120-5
0.55-150-5
235.8510.08
0.19166-55
0.67
131.8910.02
particularly demonstrating robust superiority in OOD predictions, as indicated by the lower mean (μ) and standard deviation (r) of the relative error distribution. In contrast, traditional astrostatistical methods such power-law fitting (PLF) and DCF exhibit large error values, while U-Net and DDPMs suffer from more severe performance degradation under OOD conditions. In particular, the training cost of Astro-DSB only 25% of the previous SOTA conditional DDPMs method.
Fig. further highlights the superior generalization of AStro-DSB, with sharply peaked PDF curves and coherent residual eтror patterns, especially in OOD scenarios. In comparison, both machine learning methods, U-Net modeling and cDDPMs, tend underpredict in OOD testing, while conventional methods, including 2PL and 3PL, usually overpredict for density estimations in unseen simulations. These results demonstrate that DSB offers robust and efficient predictions for astrophysical inverse tasks. Extended evaluations can be found in the appendices.
Extended robustness on observations from Taurus B213. We further test our Astro-Doo DDB on the Herschel column density map of Taurus B213 (Li and Goldsmith| 2012] Palmeirim et al. 2013) using the model trained purely on synthetic simulations. Fig. 4 visualizes the observed column density map alongside our predicted volume density map. Despite the lack of pixel-level ground truth in real-world observations, our predictions demonstrate coherent spatial structure and effectively capture the filamentary features of B213, as qualitatively verified by astrophysicists. Notably, the results align well with fangle-oinnting constraints derived from indirect measurements using the density tracer HC,N (Li and Goldsmith 2012), which suggests that Astro-DSB can generalize effectively beyond simulated scenarios, providing reliable physical predictions even on real observational data. More experimental results are in the appendices.

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Table 3: Ablation results for the key methodology designs on density predictions.
ID
OOD
Method
P-PPlle (L))
Popaa (LD
ME
M--Pele (LL)
Posla (L)
MB
w/o MI & M2
0.1943E4
1,75+0.00
66.54+1.49
4.24+2h:3
14.71=0.12
1086.66-0.66
w/o MI
-3E-377-5
0.65:6E-4
35.8620.44
-1884=5E-4
5,06444-3
-2297,48=11110
w/o M2
7E-3+1E-4
1.63-5E-
37.15+1.53
-14.666155-3
54.9513447
-12069.4499.4
Astro-DSB (Full)
-0.0211E-4
0
32.0910.665
0.51=2E-4
2.32+33-33
-5.0411.332
Observation: Herschel column density map of Tauns B213
Prediction: Number (Volume) density of Astro-DS
80
Figure 4: Visualization of observables and our predicted results on Herschel column density map of Taurus B213, shown in the plane of the sky with the x-axis representing right ascension and the y-axis representing declination. The observable map is measured in Nu/ /cm 2 and the prediction is in nı /em While there is ground truth for real-world observations, our prediction results are consistent with those obtained with other independent approaches.
Ablation studies on key methodological components. We evaluate the effectiveness of two key components our Astro-DSB design: MI: Noise alignment and M2: Observable enhancement, by systematically removing them and reporting the results in Tab. Our findings reveal that the absence of either module significantly degrades performance, with M2 being particularly effective for cases of OOD and stable prediction indicated indicated by values. The full Astro-DSS -SBo achieves the best balance of prediction accuracy and generalization robustness.
Conclusion and Discussion
In this work, address class of astrostatistical inverse prediction tasks by introducingAAtrro-DSS dynamic generative modeling framework based on the Diffusion Schrödinger Bridge. Targeting key physical quantities within GMCs, Astro-DSB learns the probabilistic transition between partial observables and true physical states, offering more flexible alternative to conventional one-step or purely discriminative approaches. Our study reveals that, in contrast to statie tasks where dynamic modeling may appear excessive, explicitly learning distributional transitions is both meaningful and beneficial in physically grounded systems demonstrated via improved generalization in oft-o--domain simulations and real-world observables. Overall, Astro-DSB offers better alignment with physical interpretability and supports the broader potential probabilistic modeling scientific discovery.
Limitation and future direction. Despite the improved robustness and performance to unseen simulation conditions, our method does not explicitly model the full forward physical equations that govern the GMC evolution. While this is reasonable given the intractable nature of observational inversion problems this work, future extensions could involve integrating multiple intermediate states, leveraging richer supervision signals, or directly encoding the dynamic physical laws into the generative modeling process to further strengthen the alignment between ML and physical dynamics for extended applications with well-defined physical forward processes.
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Acknowledgments and Disclosure of Funding
This research was partially supported by an Amazon Research Award to OR and YZ. Any opinions. findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of Amazon. DX acknowledges the support of the Natural Sciences and Engineering Research Council of Canada (NSERC), [funding reference number 568580]. DX also acknowledges support from the Eric and Wendy Schmidt Al in Science Postdoctoral Fellowship Program, a program of Schmidt Sciences. YZ also acknowledges the travel funding support by the French National Research Agency (ANR) via the "GraspGNNN" JCJC grant coordinated by Johannes F. Lutzeyer from École Polytechnique.
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Technical Appendices and Supplementary Material
A statement of broader impacts is included in Section A
Section B provides additional details on the relevant astrophysical background for readers interested in research around Giant Molecular Clouds (GMCs) and star formation. Further experimental details, including extended qualitative comparisons across different methods, are presented in Section We have also included our core code as part of the supplementary material, with the full codebase and datasets to be released upon acceptance.
A Statement of broader impacts
We apply the Diffusion Schrödinger Bridge framework to observational inverse prediction problems in astronomy. We do not anticipate any immediate societal risks or ethical concerns associated with this research. Our work focuses on advancing scientific understanding in domain with minimal direct human impact, and all data used this study are derived from publicly available astronomical observations licensed simulations.
Beyond its immediate scientific goals, this research contributes positively to the broader advancement of interdisciplinary science. By demonstrating the application of principled generative modeling framework in a physically grounded domain, our work helps bridge the gap between data-driven machine learning and theory-driven physical modeling. This approach not only promotes inter- pretability and robustness in ML research but also supports the long-term vision of using Al to accelerate scientific discovery. We hope this study encourages further collaboration between the machine learning and astrophysics communities, especially in leveraging modern ML techniques to understand complex natural systems.
B Astrophysical background
We provide extended details about GMCs and star formation research for readers interested in the astrophysical background related to this work. In this work, we focus on studying the physical conditions and processes within giant molecular clouds (GMCs), by inferring the precise distributions of various critical physical quantities from limited observational data.
Giant Molecular Clouds (GMCs) are key structures within the interstellar medium (ISM) of galaxies, composed primarily of cold gas and dust. The ISM, which fills the space between stars, also contains cosmic rays and magnetic fields and plays central role in galactic evolution by regulating star formation rates (Spitzer Jr. 2008). GMCs are of particular interest because they harbor most of the dense gas in the ISM, which is essential for the formation of stars (McKee and Ostriker 2007). The physical conditions within GMCs are highly variable and complex, exhibiting spatial and temporal variations in density, temperature, velocity, and magnetic field strength and orientation. These factors give rise to a wide array of astrophysical phenomena, including the formation of protostars (Heyer) and Dame 2015), star clusters (Krumholz al., 2019), and complex organic molecules (Bisbas et al. 2023). Among all physical quantities, we study two fundamental quantities in this work, which are the gas density (p) and the magnetic field (B).
B.1 Gas density
The gas density (i.e., mass per unit volume), often measured as the number density of hydrogen nuclei (H), is considered one of the most fundamental quantities in GMC studies. Under the assumption an abundance of one He nucleus for every 10 H nuclei in interstellar gas, we have a mass per H nucleus of μи = 1.4mH = 2.34 x 10 10-244, g. leading to пи = 1em -3 cquivalent to p = 2.34 x 10 24 gmm"" -3
The gas density directly governs gravitational collapse, sets the local timescale for star formation, and strongly influences the thermal and chemical evolution the gas. For example, denser regions within GMCs are more likely to become gravitationally unstable, leading to the formation of stars or star clusters. Moreover, the density controls the rates of radiative cooling and chemical reactions, such as molecule formation and dust grain processes, which are crucial for understanding the lifecycle of interstellar gas and the conditions for planet formation. Accurately inferring the spatial distribution of density within GMCs is therefore essential for modeling the early stages of stellar evolution.
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In astronomical studies, it is usually very challenging to precisely quantify the number density of GMCs from observations. The traditional astrostatistical approach relies on the observations of column density and certain assumptions on the geometry of the clouds. For example, a cylindrical geometry for filamentary structures or spherical geometry for dense cores (Palmeirim et al. (013).
Bisbas et al. [2021] 2023) later propose an empirical power law to convert the observed column density to the mean number density of GMCs based on MHD simulations (Wu et al. 2015). which has also been adapted in our work two baseline methods (i.e., two-component power-law conversion and three-component power-law conversion). Despite being effective in similar simulation data, the prediction performance of these methods often suffers from a significant drop when new observations present a large distributional shift compared to the original fitting data (i.e., OOD testing cases). In our work, we aim to introduce machine learning based approach with better robustness and generalization ability.
B.2 Magnetic field
Magnetic fields, often measured in microgauss (μG), represent another critical physical quantity within the interstellar medium (ISM), playing a fundamental role in shaping the structure and regulat- ing the dynamics of GMCs (Crutcher T999 2012). They can provide support against gravitational collapse, guide anisotropic gas flows, and influence the morphology of star-forming regions through magnetic tension and pressure. In combination with turbulence, magnetic fields are believed to significantly affect the efficiency and fragmentation of star formation across spatial scales.
Similar to the previous density estimation, estimating the magnetic field strength within GMCs is challenging task. The Davis Saass--aaaaaaaaaaa--aa Fermi (DCF) method is a widely adopted technique in astrophysics to estimate the plane-of-sky -ky magnetic field strength Bpos in molecular clouds based on observed dust polarization (Davis) 1951 Chandrasekhar and Fermi] 1953 Beck 2016). which relates the magnetic field strength to the polarization angle and other observable properties. Its core assumption is that an equipartition exists between the magnetic field energy and the turbulent kinetic energy of the gas. Specifically, the DCF method states that the relation between the gas density p. the nonthermal velocity dispersion ov, and the polarization angle dispersion OPA gives the POS magnetic field strength Pos as:
BPos = S ATp ov σрΛ
where / is correction factor accounting for projection and geometric effects. More recently, variant of DCF method (Skalidis and Tassis 202T) takes the compressible modes into account and modifies the previous Eq. 7 to:
Bpos = 2Tp ov
(8)
VOPA
While widely used, this method suffers from several limitations: (1) the equipartition assumption may not hold in strongly magnetized or shock-dominated regions; (2) σg becomes unreliable in regions with low signal-to-noise ratio or complex field morphology; and (3) accurate density and turbulence estimates are required for robust magnetic field inference, which are themselves uncertain. Moreover, DCF provides only an average value over large regions and lacks the ability to resolve fine-scale spatial structures.
In this work, we aim to bypass the limitations of traditional analytic approximations by leveraging machine learning directly infer the underlying magnetic field distribution from observables, jointly with other key physical quantities such as gas density. Our proposed framework thus offers a more flexible and data-driven solution, especially under conditions with limited observability or strong distribution shifts.
In Fig. 5 visualize example of molecular cloud data used in machine learning methods from Xu et al. (005)), including the map of column density with magnetic field directions, the magnetic field angle dispersion, the LOS velocity dispersion, and the projected magnetic field strength.
C Additional experimental results and analysis
We provide extended details about the experiments in this section.
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Figure 5: Illustration of different synthetic observables from (Xu et al. 2025) for density and magnetic field estimation. Top left: map column density with magnetic field directions; Top right: magnetic field angle dispersion; Bottom left: LOS velocity dispersion; Bottom right: projected magnetic field strength.
C.1 More details about the magnetolydrodynamies data simulations
We follow the same simulation setups used in prior works that apply conditional DDPMs astro- physical observational prediction (Xu et al. 2023b 2025 Wu et al. 2020 Hsu et al. 2023). While the design and analysis of physical simulations represent important research direction own that is distinct from the machine learning focus of this work, we provide only the essential background here and refer interested readers the original astrophysical literature for further details.
their
Density simulation. For the density simulations, we adopt the magnetohydrodynamic (MHD) simulations of colliding and non-colliding GMCs following prior works (Wu et al. 22200) Hsu et al.) 2023), performed using the adaptive mesh refinement (AMR) code Enzo (Bryan et al.) 2014). The simulations include self-gravity, magnetic fields, and thermal processes under photodissociation radiation field (Go = 4 Habings) and cosmic-ray ionization rate ( = 10 16 -1 Two clouds with radii of 20 pe initialized in a 128 pe domain with 2563 grid resolution.
Each cloud starts with FH 83 -3 and T 15 K, while the ambient gas set ny 8.3 cm -3 and T = 150 K to maintain pressure balance. A multiphase temperature structure develops over time, with high-density regions (n 2 10 3 cooling to 10-20 K, and low-density regions (nu X 10 cm 3) reaching up to 1000 K. Magnetic fields are initialized at 10-50 μG, oriented 60 to the collision axis. The refinement strategy ensures that the local Jeans length resolved by least eight cells.
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The simulation runs span 5 Myr, with snapshots taken at 3 and 4 Myr. Column density maps and corresponding line-of-sight (LOS) mass-weighted number density maps are extracted at multiple physical scales (32, 16, 8, and 4 pe), and projected to 128 x 128 pixel images. The resulting dataset contains 7179 samples, with an approximate 80/20 split between training and testing. For more
details, please refer to (Wu et al. 2020 Hsu et al. 2023).
Magnetic field simulation. For magnetic field prediction tasks, we adopt ideal magnetohydrody- namics (MHD) simulations following the setup in prior work (Wu et al., 2020 Hsu et al., )
using the MUSCL-Dedner method and HLLD Riemann solver within the AMR code ENZO (Bryan et al. 2014). These simulations incorporate self-gravity, magnetic fields, and thermal processes under a photodissociation radiation field with G2 = and a cosmic-ray ionization rate of < 10-16 s- Each simulation initializes two clouds (radius 20 pc) in a 128 pc domain at 256 resolution, with initial hydrogen number density Hn = 83 cm -3 temperature T = 15 K, and solenoidal turbulence satisfying of xc k- for 2 < k S 20. The magnetic field is initialized at 60" angle to the collision axis with strengths of 10, 30, and 50 JG across different cases, and four additional refinement levels are used to resolve the local Jeans length. For each magnetic field configuration, we model both colliding and non-collionn collidii GMC scenarios, where colliding clouds are offset by 0.5RGMC and have relative velocity of 10 km/s. These simulations run up 4.1 Myr and capture early evolutionary phases before star formation begins.
To generate realistic observables, we compute maps of column density, line-of-sight (LOS) mass- weighted polarization angle, LOS nonthermal velocity dispersion, and projected 3D magnetic field strength across varying AMR levels. The final dataset consists of 25,479 images at 128 128 resolution, sampled across physical scales of 32, 16, 8, and 4 pc. To enhance diversity, we include multiple projection angles and aggregate LOS effects without manually disentangling contributions such as turbulence or cloud-scale motions, consistent with observational limitations.
C.2 Additional training details
As we have mentioned in the main paper, our proposed Astro-DSB benefits from a faster convergence speed and thus less computational resources compared to conditional DDPMs and vanilla DSB models. In Fig. 6 we compare the training curves among different variants of DSB as additional ablation analysis.
Compared to previous works that leverage conditional DDPMs as the prediction framework (Xu al., 2023a 2025), our Astro-DSB converges around 25k iterations as shown in Fig. (), which corresponds to approximately 10 GPU hours. In contrast, conditional DDPMs take around 40 GPU hours to train.
(a) Training curve of the vanilla DSB
(b) Training curve the Astin-DSB wio M2.
(c) Training curve of the Astro-DSB w/o ML.
(d) Training curve
curve ofe full Astro-DBB.
Figure 6: Training curve comparison of different variants of DSB for ablation analysis. We show that the proposed M2 component on the observable enhancement contributes to faster training convergence speed. Best viewed in color and with zoo
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cDDPMs
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Figure 7: Additional visualization of relative error patterns from different machine learning methods for the density prediction task OOD testing cases. In the first row, qualitatively show the GT and prediction results from conditional DDPMs, U-Net, and our proposed Asro-DSB method. The second and third rows present the relative errors computed by "GT pred" in absolute values and the values after log transformation, respectively. Compared to other ML methods that tend underpredict OOD testing cases, our Astro-DSB achieves less biased and more robust prediction performance.
C.3 More details about experiments on simulation data
As explained in the main paper, evaluations in astrophysical prediction particularly emphasize the relative error patterns exhibited between the prediction results and the ground-truth distributions. In Fig. 7 present additional visualization of relative error patterns from different machine learning methods for the density prediction task on OOD testing cases. The first row shows the ground truth and prediction results from conditional DDPMs, U-Net, and our proposed Astro-DSB. The second and third rows visualize the corresponding relative errors, computed as "GT Pred" in absolute values and after applying log transformation, respectively. Compared DDPMs and U-Net, which tend to significantly underpredict dense structures in the OOD regime, Astro-DSB produces more balanced and spatially consistent residuals. These visual results further support the claim that method achieves improved robustness and lower bias when generalizing beyond training distributions.
To better understand the impact of each design component, we compare the distribution of relative errors for different methods in Fig. 8 Traditional power-law baselines (p2, p3) and vanilla DDPMs (cdm) exhibit large biases and heavy tails, especially under distribution shifts. The vanilla DSB (psbvl) !) slightly reduces eтrоr spread, while our proposed components- -1 and M2-each contribute to sharpening and centering the PDF. The full model (psby4) demonstrates the narrowest and most symmetric emoor distribution, highlighting its robustness and well-calibrated behavior in OOD settings.
C.4 More details about experiments on real observations
further compare our proposed Astro DSB with previous machine learning methods on real observational data from the Taurus B213 region, focusing the inferred number density distribution. As shown in Fig. 4 the prediction Astro-DSB closely matches the spatial structures and filament morphology present in the original Herschel column density map. In contrast, the prediction from
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Figure 8: PDF comparisons across different methods for the density prediction task ID and OOD data. Methods p2 and p3 correspond to two- and three-component power-law conversions, respectively. cdm denotes conditional DDPMs. psbv/ is the vanilla diffusion Schrödinger bridge without our proposed modules. pshv2 and psbu3 remove M1 (noise alignment) and M2 (observable enhancement), respectively, while psbv4 represents the full Astro-DSD model.
CASI-2D (Xu et al. 2020a) fails to capture fine-scale density variation and underestimates extended dense regions. The conditional DDPMs produce more coherent predictions but still smooth out localized dense structures. Overall, Astro-DSB exhibits better spatial continuity, structure recovery, and dynamic range, demonstrating its superior robustness and generalization capability real-world astrophysical observations.
Obscrvation: Herschel column donsity map Taurus B213
Prodiction: Number (Volume) density map of CASI-2D
1b
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Prodiction: Number (Volume) donsity map cDDPMs
Prediction: Number (Volume) densaty map Astro-DSB
Figure 9: Additional qualitative comparisons among different machine learning methods on Taurus B213 number density prediction. CASI-2D (Xu et al. 2020a) is previous ML method with U-Net architecture trained with discriminative reconstruction loss.
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Reviews:
[{"review_id": "xgQciXMNj8", "reviewer": "Reviewer_RTRd", "summary": "This paper proposes a new method called Astro-DSB with a pairwise domain assumption for solving the problem of observational inverse prediction in giant molecular clouds related to star formation. Through physical simulations and real Taurus B213 observations, it shows the method enhances interpretability, learning efficiency, and prediction performance over traditional astrostatistical and machine learning approaches.", "questions": "1. Could analytic or learned physical drifts be inserted into the bridge to improve interpretability and extrapolation?\n2. What exactly delivers the four-fold training-time speed-up over the cDDPM baseline (shorter schedule, faster convergence, architectural changes)?\n3. Do multiple stochastic samples yield meaningful uncertainty estimates for astronomers, or is the output effectively deterministic?", "limitations": "1. The dynamics are purely data-driven. Conservation laws and intermediate physical states are not enforced.\n2. Iterative sampling can be slow for large-scale deployment without further acceleration or distillation.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths:\n\nOriginality: It innovatively adapts DSB models to astrophysical dynamics, introducing the Astro-DSB framework with pairwise domain assumptions—a novel extension beyond traditional visual synthesis applications. It also introduces noise-aligned conditioning and step-wise observation injection.\n\nQuality: The research is rigorously supported by comprehensive experiments on both simulated magnetohydrodynamic data and real Taurus B213 observations. Quantitative results highlight Astro-DSB’s superior performance (e.g., reduced relative error μ=0.51±2E-4 in OOD density prediction) and 75% faster training than conditional DDPMs, validated through ablation studies and comparative analyses against astrostatistical/ML baselines.\n\nClarity: The paper is systematically structured, clearly outlining the problem setup, technical designs and evaluation metrics. Pictures and tables effectively communicate results.\n\nSignificance: The work shows that learning a full distributional trajectory can deliver stronger OOD robustness and lower training cost than conventional conditional diffusion or direct regression, with clear impact for scientific machine learning.\n\nWeaknesses:\n\n1. Incomplete physical modeling: The Astro-DSB framework does not explicitly encode the full forward physical equations governing GMC evolution. This limits its interpretability as a physics-driven model, as the learned dynamics may not fully align with underlying astrophysical laws. \n2. Limited OOD scenario validation: While OOD testing uses different MHD codes (ENZO vs. ORION2), the study focuses on specific initial conditions. It lacks validation across broader physical parameter spaces.", "was_revised": true, "final_justification": "Overall, I appreciate the authors’ careful and detailed engagement with the previous comments. The authors thoughtfully position their work not as a final solution, but as a meaningful step forward in building machine learning models with stronger physical alignment for astrophysical inverse problems, acknowledging the real challenges of encoding fundamental physics. The justification for the OOD evaluation is strong by emphasizing different physical assumptions and the test scenarios are not arbitrary, but scientifically relevant. Their choice of a data-driven approach is well-argued, especially given the absence of analytical solutions in this domain, and it reflects a principled approach to tackling complex inverse problems. Regarding uncertainty, since the outputs are deterministic, they’ve validated consistent results across multiple runs, which adds credibility.", "rebuttal": "We appreciate the valuable feedback from the reviewer. We’re also grateful for the reviewer’s recognition of the paper’s strengths, including the novel adaptation of DSBs to astrophysical dynamics, the strong OOD performance, and the significantly faster training compared to conditional DDPMs, highlighting both the methodological contribution and practical impact in scientific ML settings. \n\n\nBelow, we provide our detailed responses to the raised concerns and questions, particularly emphasizing how our proposed method contributes to *“better physical alignment over existing literature in astrophysics and other ML methods” under the intractable nature of observational inverse prediction problems*.\n\n\n---\n**W: Incomplete physical modeling as the learned dynamics may not fully align with underlying astrophysical laws.**\n\n\n**A1:** While we understand the reviewer’s concern, we would like to respectfully clarify that fully encoding the complete set of physical laws governing GMC evolution remains inherently difficult. It is a broader goal that requires sustained, collaborative efforts from the entire community and is unlikely to be fully addressed by a single paper. \n\n\nWe posit ourselves as a step forward with a better alignment ML formulation **compared to existing available ML and conventional astro methods** tackling this line of problems (see our A3 below for details), rather than a framework that fully addresses this long-term challenge. As we also noted in the Limitations section (L332–338), this difficulty stems largely from the **ill-posed and intractable nature** of inverse prediction tasks in astrophysics. In fact, even for conventional astrophysical approaches, such as the DCF baseline (Davis 1951; Chandrasekhar & Fermi 1953) included in our comparison, are **statistical methods rather than exact physical solvers**, which often rely on simplified physical assumptions (L95-101). \n\n\n---\n**W: OOD testing using different MHD codes only focuses on specific initial conditions.**\n\n\n**A2:** **Our OOD scenarios differ in more than just initial conditions.** As explained in L218–223, ENZO (ID) and ORION2 (OOD) adopt different physical assumptions: ENZO includes self-gravity, magnetic fields, radiative heating and cooling, while ORION2 excludes gravity but includes driven turbulence. These differences lead to distinct simulation behaviors and observables, which also motivate us to use them as OOD test cases. Additionally, we believe that the real observational data also serve as strong OOD testing cases.\n\n\n---\n**Q: Could analytic or learned physical drifts be inserted into the bridge to improve interpretability and extrapolation?**\n\n\n**A3:** We appreciate the insightful question. For observational inverse prediction problems, there are no known analytical physical descriptions, as these tasks **are inherently intractable**. This is also why conventional astrophysical methods we compared, such as the DCF approach (Davis 1951; Chandrasekhar & Fermi 1953), **are fundamentally statistical** rather than governed by explicit physical laws. While the forward physical evolution $\\mathcal{H}$ can be simulated from initial conditions, there are no explicit physical constraints defining the mapping $\\mathcal{F}$ between the true physical states and the observables (as discussed in L115–125). \n\n\nThis ill-posed *physical nature makes data-driven machine learning approaches particularly suitable* for tackling such astro problems. This is also the intuition behind our approach: by aligning ML and physical states through a data-driven bridge, we aim to implicitly capture the underlying dynamics. The ultimate hypothesis is that if the model can consistently align distributions of physical states across varying observations, it increases our confidence that the learned process reflects meaningful astrophysical evolution, since these states are ultimately governed by physical laws (equations). But again, this remains a long-term research goal. That said, we believe our work represents a meaningful step forward compared to existing approaches in both astrophysics and machine learning (see also A4 for further discussion).\n\n\n---\n**Q: What exactly delivers the four-fold training-time speed-up over the cDDPM baseline?**\n\n\n**A4:** The speed-up primarily stems from **the reduced distributional gap in our formulation**. In cDDPM, the model learns to map from a standard Gaussian to the physical state $x_1$, which involves a large distributional distance. In contrast, Astro-DSB learns a bridge between the observation $y$ and the target $x_1$, which are more closely aligned. This smaller gap leads to more efficient training and faster convergence. Similar trends with comparison to DDPMs have also been reported in other works such as [1].\n\n\nThis further demonstrates the advantage of our formulation in aligning with physical states, since the Gaussian prior in cDDPM lacks physical meaning (as explained in L150–158, \"...conditional DDPMs introduce unnecessary Gaussian prior assumptions...\"), whereas our method is closer to a real astrophysical observational scenario.\n\n\n---\n**Q: Do multiple stochastic samples yield meaningful uncertainty estimates for astronomers, or is the output effectively deterministic?**\n\n\n**A5:** Thank you for the question. In the physical context, particularly for the simulation data, the observable $y$ is **theoretically deterministic** given the physical state $x_1$ under a pre-defined observational process. Therefore, while uncertainty estimation is indeed an important research direction in many scientific modeling, this specific task setup *does not explicitly require it*.\n\nFrom the ML perspective, we conducted several trials of stochastic sampling using different random seeds and observed minimal variation across runs (on the order of $10^{-3}$ to $10^{-4}$, negligible in astro studies), as reported in our tables. This is consistent with prior literatures that use cDDPMs as the framework [2,3].\n\n---\n**Iterative sampling can be slow for large-scale deployment without further acceleration or distillation.**\n\n\n**A6:** Thank you for the comment. In our original experiments, we also tested the DDIM sampling strategy with skipped steps during inference, as mentioned in L287–289. We observed that this acceleration **did not lead to significant performance degradation** in our scientific prediction tasks. This suggests that faster sampling is *indeed feasible in practice*. That said, we agree that further exploration of advanced acceleration techniques, such as distillation or adaptive schedulers, remains a valuable direction for improving deployment efficiency in large-scale scientific applications.\n\n---\n[1] Physics-aligned field reconstruction with diffusion bridge, ICLR 2025\n\n\n[2] Denoising Diffusion Probabilistic Models to Predict the Density of Molecular Clouds, APJ 2023\n\n\n[3] Exploring Magnetic Fields in Molecular Clouds through Denoising Diffusion Probabilistic Models, APJ 2025"}, {"review_id": "DtiCTWetDT", "reviewer": "Reviewer_8Wk8", "summary": "This paper applies Diffusion Schrödinger Bridge (DSB) models to astrophysical systems, focusing on Giant Molecular Clouds (GMC).", "questions": "- Could the authors clarify how the cDDPMs were implemented? Was the prior simply replaced with a Gaussian, or were other modifications made?\n\n- In Equation (5), it seems there is a typo: the RHS should likely use $x_1$ rather than $x_t$.\n\n- In Equation (6), the loss term should be written as a squared norm.", "limitations": "Please refer to the points above on the lack of methodological novelty and limited depth of analysis.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 1, "originality": 1, "strengths_and_weaknesses": "- The methodology is technically reasonable and follows established frameworks for DSB-based modeling.\n\n- However, the contribution appears quite incremental. The training procedure closely mirrors existing approaches such as I2SB and Bridge-TTS, and the inference relies on a standard aggregation technique. I do not see a clear methodological innovation that sets this work apart from prior DSB literature.\n\n- Additionally, I believe the analysis is weak. While the results include robustness and generalization of the proposed method, there is no discussions or insights into why the proposed method performs better than baselines, or what aspects of the astrophysical context might benefit specifically from this approach. Additionally, I don't see the clear improvements compared to cDDPM,\n\n- The explanation of the comparisons are missing, both in the main text and appendix.  Especially, I do not see how cDDPM is implemented.", "was_revised": true, "final_justification": "In the initial review phase, I felt that the overall motivation and methodology closely followed I2SB, and the experimental results appeared unconvincing, particularly since the proposed method showed performance comparable to cDDPM in Tables 1 and 2, and the ablation studies in Table 3 did not consistently outperform baseline metrics. However, I now better appreciate the novelty of this work, especially the authors’ clarification regarding the significance of the OOD generalization results. The emphasis on out-of-distribution performance has helped clarify the contribution. I now recognize that the proposed modifications in M1 and M2 play a meaningful role in balancing fidelity and generalization, which I believe strengthens the paper’s overall contribution.", "rebuttal": "We appreciate the comments from the reviewer, below we provide our detailed responses to raised questions.\n\n---\n**No discussions or insights into why the proposed method performs better than baselines, or what aspects of the astrophysical context might benefit specifically from this approach. Additionally, I don't see the clear improvements compared to cDDPM.**\n\n**A1:** All the requested discussions have been **comprehensively included in the original submission**.\n\n- **A.1.1:** Insights into why the proposed method performs better than baselines.\n\n   - **For astrostatistical baselines, L95-101:** “*Traditionally in astronomy, such observational inverse prediction problems have been commonly approached via astrostatistical methods, which often combine theoretical simplifications with statistical priors…*” In other words, due to the intractable nature of observational inverse prediction tasks, conventional astro methods, such as the DCF approach (Davis 1951; Chandrasekhar & Fermi 1953), are fundamentally statistical rather than governed by explicit physical laws. Our proposed data-driven ML approach offers advantages by learning complex nonlinear mappings without relying on oversimplified assumptions.\n\n   - **For existing ML baselines, particularly recent cDDPMs, L150-158:** “*... conditional DDPMs introduce unnecessary Gaussian prior assumptions, which are not naturally aligned with the physics of GMCs. Our proposed Schrödinger bridge framework removes this assumption by directly modeling the transition between observable $y$ and the true physical distribution $x_1$, resulting in a distribution-level probabilistic mapping without reliance on artificial prior structures, thus offering better ML interpretability compared to conditional DDPMs.*”\n\n- **A.1.2:** Aspects of the astrophysical context might benefit specifically from this approach.\n\n   - As described in L150–158, the proposed formulation enables state-to-state alignment between the observed data and the underlying physical states, which provides an effective inductive bias for learning meaningful dynamics. Moreover, given the intractable and statistical nature of inverse prediction tasks, the advantage of our method lies in bypassing the need for artificial priors (like Gaussian noise) and instead leveraging observable-informed transitions to achieve better dynamic fidelity and generalization.\n\n- **A.1.3:** Improvements compared to cDDPM.\n\n   - **Performance-wise, Tab.1, Tab.2 and Fig. 3, L291-297**: “*...particularly demonstrating robust superiority in OOD predictions, as indicated by the lower mean (µ) and standard deviation (σ) of the relative error distribution…*”. For instance, in Table 2, our ID density prediction achieves a mean error $\\mu$ of -0.02, compared to -0.05 from cDDPM. In the OOD case, the relative error is 0.51 for Astro-DSB, compared to 2.88 for cDDPMs.\n\n   - **Efficiency-wise, L297-298, Appendix C.2:** “*In particular, the training cost of Astro-DSB is only 25% of the previous SOTA conditional DDPMs method.*” Figure 6 in Appendix C.2 further shows the faster convergence of Astro-DSB during training.\n\n---\n**Could the authors clarify how the cDDPMs were implemented? Was the prior simply replaced with a Gaussian, or were other modifications made?**\n\n**A2:** The cDDPM baselines are introduced in prior literature [1, 2], and we followed the publicly available codebases for reproduction in our paper. This includes the standard conditional DDPM framework with 1000 diffusion steps, where the observables $y$ are passed as conditioning inputs $c$ into the U-Net at each step during training to model $p(x_{t-1}|x_t, c)$. No explicit aggregate sampling techniques were used.\n\nAt a high level, one core difference between cDDPMs and DSB formulation lies in the starting distribution: cDDPMs rely on a fixed Gaussian prior, whereas DSB directly models a bridge between the observation $y$ and the target distribution $x_1$. As explained in L150–158, this difference enables better state alignment and contributes to significantly faster convergence (25% of the cDDPM training), as the distributional distance between the two endpoints is smaller and more physically meaningful.\n\nIn summary, the cDDPM implementations are directly based on prior literature, but we would be happy to clarify this further or include additional implementation details in the revised version if the reviewer finds it helpful.\n\n---\n**In Eq. (6), it seems there is a typo: the RHS should likely use $x_1$  rather than $x_t$. In Eq. (7),  the loss term should be written as a squared norm.**\n\n**A3:** We have carefully reviewed Eq. (6) and Eq. (7) and confirm that there are no typos. In Eq. (6), our intention is to present the posterior distribution of the intermediate state $x_t$ conditioned on $x_1$, which is consistent with the formulation. As for Eq. (7), we use the L1 norm rather than the squared L2 norm, and this has been a rather standard operation as in prior literature [3].\n\n---\n**Contribution appears quite incremental.**\n\n**A4:** We understand the reviewer’s concern, however, we would like to respectfully clarify that this work is not merely a methodology paper, and the vanilla I2SB framework is not directly applicable in these inverse tasks.\n\nFirst, our proposed method demonstrates **consistent improvements in both prediction accuracy and training efficiency** compared to both conventional astrostatistical approaches and recent ML baselines. As stated in the abstract: “*From the astrophysical perspective, our proposed paired DSB method improves interpretability, learning efficiency, and prediction performance over conventional astrostatistical and other machine learning methods.*”\n\nSecond, we emphasize the generalization ability of the proposed method under **physically motivated out-of-distribution (OOD) settings and validate its robustness on real astronomical observations (Taurus B213)**, which are rarely addressed in standard ML contexts. This constitutes a non-trivial and essential step toward applying machine learning methods in scientific domains where interpretability, robustness, and physical grounding are crucial.\n\nFinally, the vanilla I2SB framework is not **directly applicable to inverse prediction problems**. Originally designed for image translation, it corresponds to our setting without modules M1 and M2. As shown in our ablation studies, both components are essential for achieving strong ID and OOD performance.\n\nIn this sense, our work contributes not only methodologically, but also meaningfully bridges machine learning and astrophysical research through practical and verifiable advancements.\n\n---\n[1] Denoising Diffusion Probabilistic Models to Predict the Density of Molecular Clouds, APJ 2023\n\n[2] Exploring Magnetic Fields in Molecular Clouds through Denoising Diffusion Probabilistic Models, APJ 2025\n\n[3] Physics-aligned field reconstruction with diffusion bridge, ICLR 2025"}, {"review_id": "ETTcFyXvuX", "reviewer": "Reviewer_Ey1R", "summary": "This paper introduces Astro-DSB, a specialized Diffusion Schrödinger Bridge model for astrophysical inverse prediction tasks within Giant Molecular Clouds (GMCs). The authors adapt the I2SB framework with three modifications: (M0) pairwise marginal distribution matching, (M1) noise perturbation alignment with observational uncertainties, and (M2) observable enhancement throughout the diffusion trajectory. The method is evaluated on both synthetic magnetohydrodynamics (MHD) simulations and real observational data from Taurus B213, demonstrating improved performance over conventional astrostatistical methods and other ML approaches in both in-distribution (ID) and out-of-distribution (OOD) settings.", "questions": "1. How does the learned diffusion trajectory relate to the actual physical evolution of GMCs? The paper claims to capture \"dynamic evolution processes\" but doesn't establish this connection rigorously.\n\n2.The choice of λ = 0.1 for the observational noise appears arbitrary. How sensitive are the results to this parameter, and how should it be chosen for different observational scenarios?\n\n3. How does the method scale with the number of physical quantities to predict? Would the pairwise matching assumption (M0) become limiting for multi-output scenarios?\n\n4.The comparison with conditional DDPMs uses the same training epochs (100 vs 400), but it's unclear if this represents convergence for both methods. Could the DDPM performance improve with longer training?\n\n5.What specific aspects of the DSB formulation enable better OOD generalization compared to discriminative approaches? The explanation remains largely empirical.", "limitations": "The approach is limited to paired training data, which may not always be available in real observational scenarios. please also refer to the cons and questions. thank you.", "rating": 4, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 2, "strengths_and_weaknesses": "pros\n1.The application of diffusion bridge models to astrophysical inverse problems represents a meaningful extension beyond traditional computer vision tasks, addressing scientifically important problems in star formation research.\n\n2. Unlike heuristic OOD definitions in vision tasks, this work provides principled OOD evaluation through variations in initial physical conditions and dominant physical processes, making the generalization assessment more scientifically grounded.\n\n3. The evaluation on real observational data from Taurus B213 demonstrates practical applicability beyond synthetic simulations, which is crucial for scientific applications.\n\n4.The reported 75% reduction in training time compared to conditional DDPMs while achieving superior performance is practically significant.\n\ncon s:\n1. The core technical contributions are relatively incremental modifications to existing I2SB framework. The three proposed modifications (M0, M1, M2) are straightforward adaptations rather than fundamental algorithmic innovations.\n\n2.The paper lacks theoretical analysis of why DSB should be particularly well-suited for astrophysical inverse problems compared to other generative approaches. The connection between the physical dynamics of GMCs and the SDE formulation is not rigorously established.\n\n3.The synthetic data comes from only two MHD codes (ENZO and ORION2), potentially limiting the generalizability of conclusions about OOD performance.", "was_revised": true, "final_justification": "as I said, pros: The authors cleared up my questions about the physical dynamics connection (Q1) and noise sensitivity (Q2). Reviewer 8Wk8 made a good point about M1/M2, the ablations show it's not just more parameters but actual modeling improvements. The ENZO vs ORION2 OOD setup is actually stronger than I first thought since they use different physics, not just initial conditions. cons: Still think the novelty over I2SB is limited, but the 75% faster training and working on real data makes it worthwhile for the astro community? So I cant give a score more than 4.", "rebuttal": "We thank the reviewer for the valuable feedback and are encouraged by the recognition of our principled OOD setup, evaluation on real observations, and the practical significance of improved training efficiency. Below we address the raised questions in detail.\n\n---\n**Q1: How does the learned diffusion trajectory relate to the actual physical evolution of GMCs?**\n\n**A1:** We appreciate the insightful question. Our interpretation of dynamics is established through explicit state alignment in a data-driven manner, which we will further elaborate on from the following perspectives:\n\n- **The nature of observational inversion problems.** As noted in L95–101 and L115–125, while the physical forward process $\\mathcal{H}$ from initial condition $x_0$ to current physical state $x_1$ is governed by known equations and simulated explicitly, the inverse prediction from observables $y$ is **fundamentally intractable**, similar to many other scientific inverse problems. As a result, even established astrophysical methods like the DCF approach (Davis 1951; Chandrasekhar & Fermi 1953) are fundamentally statistical rather than derived from physical laws. This ill-posed nature makes data-driven machine learning approaches particularly suitable for such tasks.\n\n- **How can our method better capture underlying dynamics in this case?** Despite the intractability, it is well accepted in astrophysics that *the observables $y$ are indirectly shaped by the underlying dynamics of GMC evolution*. Our assumption is that, by aligning observable $y$ and physical state $x_1$ via a data-driven bridge, we can implicitly capture this evolution. The key hypothesis is: if our model can consistently align the distributions of physical states across varying observations, we gain greater confidence that the learned transitions reflect meaningful astrophysical dynamics, **given these states are ultimately also governed by physical laws (despite they remain to be unknown yet in astro)**. Compared to SOTA cDDPMs, which rely on an artificial Gaussian prior (see L150–158), our bridge formulation avoids this assumption, resulting in more physically grounded state alignment and faster convergence.\n\nThat said, We acknowledge that directly encoding physical equations into the generative process is a longer-term goal, but believe our method offers a principled and empirically supported step **over existing ML and astrophysical approaches**.\n\n---\n**Q2: The choice of λ = 0.1 for the observational noise appears arbitrary. How sensitive are the results to it?**\n\n**A2:** Thank you for the comment. The choice of $\\lambda = 0.1$ is not arbitrary, but follows statistical conventions as we mentioned in L.180. \n\nIn **existing astro literature**, noise is commonly introduced both by the instrumentation (the telescope) and by foreground/background gas emission with a signal-to-noise ratio (SNR) at the order of 10.  For instance, the noise level from the column density map from the Five College Radio Astronomical Observatory (FCRAO) based on the 12CO observations [1] is around 0.28K, v.s. signal of 1-7K. Similarly, the extinction maps derived from Spitzer infrared observations toward several molecular clouds [2,3] shows ~10% uncertainties.\n\nIn the **ML context**, we additionally varied $\\lambda$ from 0.1 to 0.3. Overall, ID performance remains stable, while a larger $\\lambda$ slightly degraded OOD results but **still outperformed** DDPMs.\n\n| $\\lambda$ | ID-$\\mu$ ($\\downarrow$) | ID-$\\sigma$ ($\\downarrow$) |  ID-ME | OOD-$\\mu$ ($\\downarrow$) | OOD-$\\sigma$ ($\\downarrow$) | OOD-ME |\n|:--------:|:--------:|:-----------:|:------:|:---------:|:------------:|:------:|\n|    0.1   |   -0.02  |     0.73    |  32.09 |    0.51   |     2.32     |  -5.04 |\n|   0.15   |   0.03   |     1.61    | 145.04 |    2.52   |     4.91     | 313.00 |\n|    0.2   |   -0.02  |     1.58    | 131.34 |    2.16   |     5.33     | 186.37 |\n|   0.25   |   -0.02  |     1.41    |  90.65 |    2.19   |     5.28     | 164.43 |\n|    0.3   |   0.03   |     1.74    | 141.40 |    2.71   |     5.71     | 240.55 |\n\n---\n**Q3: How does the method scale with the number of physical quantities to predict? Would the pairwise matching assumption (M0) limit multi-output scenarios?**\n\n**A3:** **Physically**, M0 stems from the intrinsic nature of observational inverse prediction tasks. Specifically, under a fixed observational process $\\mathcal{H}$, the physical state $x_1$ should **produce a deterministic observation $y$**. Therefore, multi-output scenarios (i.e., multiple $x_1$ for a given $y$) should already be **encoded in the simulation data** rather than arising from the formulation ambiguity.\n\nAs for **ML scalability**, DSB actually scales better than SOTA DDPMs, which require modeling across three distributions (i.e., $y$, $x_1$, and a Gaussian prior) and introduce additional complexity and potential misalignment. In contrast, DSB avoids the artificial Gaussian prior by directly modeling the OT bridge between $y$ and $x_1$, simplifying learning and improving generalization. Similar findings have been reported in both interdisciplinary domains [4] and image generation literature [5].\n\n**Additional empirical support:** In our experiments, predicting magnetic fields (a vector field) from observations shows scalability across output dimensionality beyond scalar density. The successful results on both tasks empirically demonstrate the scalability of our method w.r.t. the dimensionality and complexity of physical quantities.\n\n---\n**Q4: The comparison with cDDPMs uses the same training epochs, but unclear if this represents convergence for both methods. Could the DDPM improve with longer training?**\n\n**A4:** We ensured that the reported training epochs correspond to convergence for each method. In Appendix C.2, Fig. 6, we also showed different training curves, and the performance of DDPMs no longer improves with longer training. \n\n---\n**Q5: What specific aspects of the DSB enable better OOD generalization compared to discriminative approaches?**\n\n**A5:** Thank you for the insightful question. The key difference in DSB lies in its distribution-level modeling between the observable and the physical state, as opposed to directly learning a discriminative point-to-point mapping. We hypothesize that this design improves OOD generalization through the following:\n\n- **Data distribution**: bridge-based generative modeling mitigates distributional mismatch. Discriminative models often struggle with OOD generalization due to substantial covariate shifts between input and output spaces. In contrast, DSB explicitly learns the stochastic evolution from $y$ to $x_1$ via a bridge, offering greater flexibility and *broader coverage over the underlying data distribution*. Under a shifted data sample, the model is more likely to operate within a plausible region of the learned data manifold, resulting in more reliable predictions.\n\n- **Physical modeling**: DSB enables explicit alignment between physically meaningful states, which encourages implicit consistency with the natural dynamics of GMC evolution. As discussed in A1, both boundary states in astro are governed by hidden physical processes, and the intermediate transitions are often treated statistically (e.g., Gaussian processes). The DSB, grounded in optimal transport, naturally captures such transitions and implicitly respects these physical assumptions.\n\n---\n**Limited technical nolvety over I2SB.**\n\n**A6:** We understand the reviewer’s concern, however, we would like to respectfully clarify that this work is **not merely a methodology** paper, and the vanilla I2SB framework **is not directly applicable** in these inverse tasks.\n\nFirst, our proposed method demonstrates **consistent improvements in both prediction accuracy and training efficiency** compared to both conventional astrostatistical approaches and recent ML SOTA baselines, as stated in our abstract.\n\nSecond, we emphasize the generalization ability under **physically motivated out-of-distribution (OOD) settings and validate its robustness on real astronomical observations (Taurus B213)**, which are rarely addressed in standard ML contexts. This constitutes a non-trivial and essential step toward applying machine learning methods in scientific domains where interpretability, robustness, and physical grounding are crucial.\n\nFinally, the vanilla I2SB framework is not **directly applicable to inverse prediction problems**. Originally designed for image translation, it corresponds to our setting without modules M1 and M2. As shown in our ablation studies, both components are essential for achieving strong ID and OOD performance.\n\nIn this sense, our work contributes not only methodologically, but also meaningfully bridges machine learning and astrophysical research through practical and verifiable advancements.\n\n---\n**Limited OOD generalization given two MHD codes (ENZO and ORION2).**\n\n**A7:** We believe that our experiments provided strong OOD generalization ability. As noted in L218–223, beyond differing initial physical conditions, ENZO and ORION2 have also **adopt different physical assumptions**: ENZO includes self-gravity, magnetic fields, radiative heating and cooling, while ORION2 excludes gravity but includes driven turbulence. These differences lead to distinct simulation behaviors and observables to ensure the robustness of OOD generalization. Additionally, our test on real observable data from Taurus B213 also supports the generalization claim.\n\n---\n[1] Large-Scale Structure of the Molecular Gas in Taurus Revealed by High Linear Dynamic Range Spectral Line Mapping, APJ 2008\n\n[2] Spectroscopic Infrared Extinction Mapping as a Probe of Grain Growth in IRDCs, APJ 2015\n\n[3] Denoising Diffusion Probabilistic Models to Predict the Density of Molecular Clouds, APJ 2023\n\n[4] Physics-aligned field reconstruction with diffusion bridge, ICLR 2025\n\n[5] Simple diffusion: End-to-end diffusion for high resolution image, ICML 2023"}, {"review_id": "7AJoAQ6rgZ", "reviewer": "Reviewer_fbj1", "summary": "In this paper, the authors propose to use a Bridge Matching procedure in order to astrostatistical inverse problems within iant Molecular Clouds. Their method is similar to I2SB [1]. The authors introduce a few key changes however which improve the methods, especially on OOD tasks. More precisely, they assume a modelling noise which is Gaussian and also further condition the model based on the clean observation $y$ similarly to [2]. Finally, as the data in astrostatistical setting is extremely large, they proceed using patch based training and an aggregation method. They conduct their evaluation on two key physical processes from magnetohydrodynamics simulations. They show improvements over conditional diffusion model baselines especially in OOD settings. \n\n[1] I^2SB: Image-to-Image Schrödinger Bridge (2023) -- Liu et al. \n\n[2] Augmented Bridge Matching (2023) -- De Bortoli et al.", "questions": "L.8 “our proposed paired DSB method” > We would like to take the opportunity to clarify that “paired DSB” is non-sensical as DSB and any Schrodinger Bridge and Optimal Transport (OT) based method must work on unpaired data as one of the goal of OT is to discover said pairing. We are aware of some confusion in the literature but we feel like this misunderstanding should be properly addressed.\n\nL.28 “we argue that the true power of this reformulation lies in the fact that it allows for diffusion models to capture the underlying data dynamics” > There are other benefits of OT compared to diffusion models especially since they work in unpaired settings as highlighted above.\n\nl.167  > I2SB is not a special case of the Schrodinger Bridge problem. \n\nL.174 > In M1, compared to I2SB the authors introduce another noise variable in $y_1 - \\varepsilon$. Does this $\\varepsilon$ correspond to the one introduced in (1), i.e., when specifying the perturbation model. It seems to be the case reading l.178. How realistic is this assumption? \n\nL.183 > The proposed method in M2 has already been proposed in the literature in [1]. The only difference between [1] and the proposed model is that [1] would take $y - \\varepsilon$ as an input while the current model takes $y$ as an input. \n\nL.189  “framework and helps simultaneously obtain lower loss values with a faster convergence speed” > It is unclear to me if the claim of lower loss value is grounded or not as a model that takes $y$ as a conditioning will have more parameters than a model that doesn’t. I guess, the main question is: “Is the improvement in performance observed by the authors only due to the increased number of parameters or is it due to the modeling choices?” My guess is that the modeling choices matter however, I do not think that the current experiments are enough to support such a claim. \n\nL.202 > I think it would be very interesting to describe in more details the aggregation strategy of AstroDSB.\n\nL.214 > I am by no means an expert in astrophysics and therefore will provide comments based on a machine learning point of view here. However, it seems that 5707 training samples in the molecular density casse and 19100 training samples in the magnetic field strength case is way less than the size of classical datasets used in machine learning. I have a few questions there: 1) is the size of the training dataset considering the patch-based augmentation? 2) since the dataset is quite small, are there safeguards in place to show that the model does not memorize the example ground-truths?\n\nL.204 > While describing the astrophysical data, even though the authors give the size of the patch it would be interesting to also give the size of the molecular density and the magnetic fied strength, i.e., how may patches do I need to reconstruct the data?\n\nL.267 > It seems that the main method of evaluation in this work is to report the mean and standard deviations of the relative errors. The authors write an interesting paragraph about the limitation of the Mean Error (ME). Is it a novel observation or is it well-known in the literature?\n\nL.274 > While investigating the implementation details it seems that the method used by the authors rely on quite outdated methods. Notably using a time discretization of 1000 steps. I would like to encourage the authors to investigate recent works such as [2,3] to improve their results.\n\nFigure 3 (b) is extremely hard to parse.\n\n[1] Augmented Bridge Matching (2023) – De Bortoli et al. \n\n[2] Simpler Diffusion (SiD2): 1.5 FID on ImageNet512 with pixel-space diffusion (2024) – Hoogeboom et al. \n\n[3] Simple ReFlow: Improved Techniques for Fast Flow Models (2024) – Kim et al.", "limitations": "The authors have explicitly addressed the limitations of their work in the conclusion.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 2, "strengths_and_weaknesses": "Strength:\n\n* The use of I2SB techniques for astrostatistical problem is new. \n\n* The results regarding the differences with conditional diffusion models are quite compelling. \n\nWeaknesses:\n\n* There seems to be some confusion around methodological and technical terms (see below)\n\n* (Minor) It is not clear to me what is the significance of those results in the astrostatistical community as I am not an expert on the topic.", "was_revised": false, "final_justification": "", "rebuttal": "We greatly appreciate the reviewer’s careful and thorough suggestions regarding the terminology clarifications. We will incorporate these writing suggestions to ensure greater clarity and rigor. Below, we provide detailed responses to the specific questions raised.\n\n---\n**L.174 -> In M1, compared to I2SB the authors introduce another noise variable in $y - \\lambda$. Does this correspond to the one introduced in (1), i.e., when specifying the perturbation model. How realistic is this assumption?**\n\n**A1:** Yes, the noise variable $\\lambda$ corresponds to $\\varepsilon$ introduced in Eq. (1), and this is a **realistic and widely adopted assumption**. In **existing astro literature**, noise is commonly introduced both by the instrumentation (the telescope) and by foreground/background gas emission with a signal-to-noise ratio (SNR) at the order of 10.  For instance, the noise level from the column density map from the Five College Radio Astronomical Observatory (FCRAO) based on the 12CO observations [1] is around 0.28K, v.s. signal of 1-7K. Similarly, the extinction maps derived from Spitzer infrared observations toward several molecular clouds [2,3] shows ~10% uncertainties.\n\n---\n**L.189 -> Is the improvement in performance observed by the authors only due to the increased number of parameters or is it due to the modeling choices?**\n\n**A2:** We appreciate the thoughtful question. To further examine whether the observed performance gain is solely due to the modeling choice, we conducted an additional ablation where the conditioning inputs were replaced with zero-padding to ensure the number of parameters remained the same. The results remain consistent with those reported in the original manuscript, supporting our claim that the improvement **stems from the modeling formulation rather than increased capacity**. We are happy to include these new results in the revised version, and provide the corresponding curve as those in Figure 6 from Appendix C.2 as rigorous validation (as we are not able to provide additional figures in the rebuttal this year).\n\n| cond | ID-$\\mu$ ($\\downarrow$) | ID-$\\sigma$ ($\\downarrow$) |  ID-ME | OOD-$\\mu$ ($\\downarrow$) | OOD-$\\sigma$ ($\\downarrow$) | OOD-ME |\n|:--------:|:--------:|:-----------:|:------:|:---------:|:------------:|:------:|\n|   ours   |   -0.02  |     0.73    |  32.09 |    0.51   |     2.32     |  -5.04 |\n|   void   |   -0.10   |     2.12    | 62.72 |    6.16   |     12.23     | 381.70 |\n\n---\n**L.202 -> Would be very interesting to describe in more detail the aggregation strategy of AstroDSB.**\n\n**A3:** The aggregation strategy in Astro-DSB is designed for compatibility with high-resolution real-world observations. Specifically, given a large observational map $y_{obs}$, we first perform a normalization step as preprocessing to convert it into a normalized input space. We then partition the normalized map into overlapping patches $y_{obs}^i$ using a fixed step size $k$, which is conceptually similar to the stride in convolutional neural networks. For example, if the patch size is $128×128$ and step $k = 8$, the top-left coordinates of successive patches shift by 8 pixels.\n\nEach patch is then independently passed through the trained model to obtain corresponding predictions $\\tilde{x}_1^i$. Finally, the overlapping predicted patches are aggregated by averaging over overlapping regions, producing a full-resolution output map aligned with the input observation. This strategy enables scalable and accurate inference on large sky surveys without introducing padding artifacts. We are happy to further clarify this process in the revised manuscript if helpful.\n\n---\n**L.214 -> 1) Is the size of the training dataset considering the patch-based augmentation? 2) Since the dataset is quite small, are there safeguards in place to show that the model does not memorize the example ground-truths?**\n\n**A4:** Thank you for the questions.\n\n- The reported training dataset size does not include the patch-based augmentation. \n\n- Regarding memorization: this is less of a concern in our setting compared to conventional ML, and does not appear to be an issue based on our OOD results.\n\n   - *Why is memorization less relevant in our inverse prediction task?* Unlike image generation tasks where memorization can lead to privacy or copyright concerns, the goal here is to recover physical quantities grounded in simulation-based truth. The focus is on whether the model can generalize the underlying mapping from observation to physical state driven by implicit physical dynamics, rather than replicate specific inputs.\n\n   - *Why are we confident the model isn’t memorizing?* Our OOD test cases are drawn from very different simulations (with distinct dominant physical processes) or real observations (Taurus B213), showing strong distributional shifts from training data. If the model were merely memorizing training examples, both discriminative ML baselines and classical astrophysical methods should perform similarly well, but the results show otherwise.\n\n---\n**L.204 -> It would be interesting to also give the size of the molecular density and the magnetic field strength, i.e., how many patches do I need to reconstruct the data?**\n\n**A5:** This question touches on key aspects of both the training data and observational setup in astronomy. As a briefsummary, the number of patches needed depends primarily on the size of the observational map and the target resolution. \n\n- **In real observations**, typical molecular clouds range from 10 to 300 light-years. For instance, the Taurus GMC spans roughly 78 light-years. The test case used in our study focuses on a specific region within the Taurus B213 filament, which is approximately 32 light-years in length.\n\n- **In practical studies**, including our ML setup, we used a physical resolution of 0.12 light-years per pixel at the distance of Taurus, which aligns well with the resolution range in our training data. It’s important to note that telescopes provide angular resolution, not physical resolution. To convert to physical resolution, we multiply the angular resolution by the distance to the cloud. As a result, more distant clouds will have coarser physical resolution, even with the same angular resolution.\n\n- **As for visualization purposes**, our Taurus B213 map shown in the paper has a pixel resolution of 128×264. Following our aggregated sampling as detailed in **A3**, we used 18 patches to reconstruct it.\n\n---\n**L.267 -> The authors write an interesting paragraph about the limitation of the Mean Error (ME). Is it a novel observation or is it well-known in the literature**\n\n**A6:** Yes, this is a rather novel observation in the ML context, yet critical in scientific domains like astrophysics where data often spans an extremely wide dynamic range (e.g., 10^{0} to 10^{7} in the case of density). This limitation of ME is less commonly discussed in standard ML and computer vision tasks, where inputs like RGB images are naturally bounded in [0, 255], and such scale imbalances are rare. In contrast, astrophysical data, particularly when unnormalized and interpreted in its physical units, exhibits significant value imbalances that can distort ME interpretation. We believe highlighting this issue is particularly relevant for applying ML to real-world scientific data.\n\n---\n**L.274 -> Encourage the authors to use more recent works, such as Simpler Diffusion (SiD2) and Simple ReFlow to improve results.**\n\n**A7:** We appreciate the suggestion and are aware of the fact that many recent ML works aim to improve image synthesis via updated training recipes, advanced schedulers, or sampler designs, including those mentioned. In our experiments, we did try alternative inference techniques with skipped diffusion steps, as noted in L.289: “We observe no significant difference in prediction performance across these sampler settings in our work.” Regarding training recipes, it seems that there is no publicly available code for SiD2 or Simple ReFlow, and thus, we were unable to reproduce them during the rebuttal phase. That said, we still thank the reviewer for the recommendation and consider these valuable directions for future exploration.\n\n---\n**On the significance of this ML work in the astrostatistical setting.** \n\n**A8:** The significance of our work can be understood from both the scientific importance of the tasks and the methodological value.\n\n- **Scientific relevance**: Understanding the physical conditions of GMCs is central to the study of star formation, which is an area of active and long-standing interest in the astrophysics community (e.g., see references McKee and Ostriker, 2007; Larson, 2003; Madau and Dickinson, 2014; Xu, 2023, 2025 and others in our original manuscript). Accurate inference of physical quantities from observational data is essential to uncovering the initial conditions that govern how stars form and evolve.\n\n- **Methodological contribution**: Traditional astrostatistical methods rely on simplified assumptions and linear approximations, while existing machine learning approaches often focus on pointwise regression or generative priors that lack physical grounding. Our proposed DSB framework improves upon both by modeling distribution-level transitions between physical states and observations. This enables better generalization across different simulation setups and robustness when applied to real observational data, as demonstrated in our OOD and Taurus B213 results.\n\n---\n**For other suggestions:** We sincerely appreciate the detailed feedback and would be happy to incorporate these suggestions into the revised manuscript.\n\n---\n[1] Large-Scale Structure of the Molecular Gas in Taurus Revealed by High Linear Dynamic Range Spectral Line Mapping, APJ 2008\n\n[2] Spectroscopic Infrared Extinction Mapping as a Probe of Grain Growth in IRDCs, APJ 2015\n\n[3] Denoising Diffusion Probabilistic Models to Predict the Density of Molecular Clouds, APJ 2023"}]

Author rebuttal:

Meta-review: The authors present Astro-DSB and apply it to a variety of applications in astrophysics. The method primarily builds on diffusion Schrödinger bridge approaches in the literature and with tailored, but fairly straightforward, modifications apply these to astrophysics. The method, Astro-DSB, is shown to perform better than or comparable to standard diffusion baselines with a significant improvement in training time.

The discussion focused on the concerns about methodological novelty, the significance of the experimental contributions, partially alleviated by the rebuttal, as well as the relevance for a machine learning conference. Ultimately, the discussion concluded that this paper is more of an application paper where the experimental validation and insights are potentially useful as a baseline and evaluation protocol for astrophysics in ML. The reviewers reached a consensus recommending weak accept for the potential utility for the astrophysics and AI4Science communities, which I don't see a strong reason to disagree with.
Decision: Accept (poster)