Physics vs Distributions: Pareto Optimal Flow Matching with Physics Constraints
Authors: Giacomo Baldan
Qiang Liu
Alberto Guardone
Nils Thuerey
Venue: ICLR 2026 Poster (2026)
Keywords: Flow Matching
Physics
PDE
Diffusion Models
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)
TL;DR: Physics-Based Flow Matching (PBFM) is a generative modeling framework that enforces physical constraints during training

Abstract: Physics-constrained generative modeling aims to produce high-dimensional samples that are both physically consistent and distributionally accurate, a task that remains challenging due to often conflicting optimization objectives. Recent advances in flow matching and diffusion models have enabled efficient generative modeling, but integrating physical constraints often degrades generative fidelity or requires costly inference-time corrections. Our work is the first to recognize the trade-off between distributional and physical accuracy. Based on the insight of inherently conflicting objectives, we introduce Physics-Based Flow Matching (PBFM) a method that enforces physical constraints at training time using conflict-free gradient updates and unrolling to mitigate Jensen's gap. Our approach avoids manual loss balancing and enables simultaneous optimization of generative and physical objectives. As a consequence, physics constraints do not impede inference performance. We benchmark our method across three representative PDE benchmarks. PBFM achieves a Pareto-optimal trade-off, competitive inference speed, and generalizes to a wide range of physics-constrained generative tasks, providing a practical tool for scientific machine learning. 

Code and datasets available at [https://github.com/tum-pbs/PBFM](https://github.com/tum-pbs/PBFM).

Published as a conference paper at ICLR 2026
PHYSICS vs DISTRIBUTIONS: PARETO OPTIMAL FLOW MATCHING WITH PHYSICS CONSTRAINTS
Giacomo Baldan
Qiang Liu
Technical University of Munich Garching, Germany
Technical University of Munich Garching, Germany Blang7.1iu@tune
giacomo. . ba dan @ tumm de
Alberto Guardone
Nils Thuerey
Politecnico di Milano
Technical University of Munich
Milan, Italy
Garching, Germany
berto. gu guttttogguaddoeepplllooo
nils a thuerry@tuum... de
ABSTRACT
generative modeling aims to produce high- digh-dimensionll sam- ples that are both physically consistent and distributionally accurate, a task that remains challenging due to often conflicting optimization objectives. Recent ad- vances in flow matching and diffusion models have enabled efficient generative modeling, but integrating physical constraints often degrades generative fidelity or requires costly inference-ting corrections. Our work the first to recognize the trade-off between distributional and physical accuracy. Based on the insight of inherently conflicting objectives, we introduce Physics-Basee Based Flow Matching (PBFM) method that enforces physical constraints at training time using conflict- free gradient updates and unrolling to mitigate Jensen's 's gap. Our approach avoids manual loss balancing and enables simultaneous optimization of generative and physical objectives. As a consequence, physics constraints do not impede infer- ence performance. We benchmark our method across three representative PDE benchmarks. PBFM achieves a Pareto-optimal trade-off, competitive inference speed, and generalizes to wide range of physics-constrained generative tasks, providing a practical tool for scientific machine learning.
Code and datasets available httptttt::// /eattss//gitii //tithub. wow com/tum pbs 2BFM
INTRODUCTION
Partial differential equations (PDEs) provide the core mathematical framework for modeling the evo- lution of physical systems across space and time (Evans) 2010). However, discretizing PDEs often results in high-dimensional problems that are computationally prohibitive, especially for nonlinear multiscale phenomena (Haber al. 2018 Valencia al. 2025). Recently, machine learning- based methods have emerged as efficient alternatives, enabling the approximation of PDE solutions with significantly reduced computational cost (Chen al., 2021 Fresca et al.) 2021 Baldan al.,
2021 Brunton & Kutz, (024).. Physics-informed neural networks (PINNs) (Raissi el al. 2019) embed PDE constraints directly into the training objective via automatic differentiation. However, PINNs yield single deterministic solution, limiting their use in scenarios where stochasticity is cru- cial, such uncertainty quantification (Roy Oberkampt] [2011] [ Abdar al.) 2001 & Thuerey 2024). In contrast, generative models like denoising diffusion probabilistic models (DDPMs) (Ho) ct al. 2020 Nichol & Dhariwal 202I), their implicit variants (DDIMs) (Song et al. (022), and flow matching (Lipman et al. 2023) have shown strong performance in capturing complex data distribu- tions across domains, including images, videos, audio, and graphs (Rombach et al. 2022 Ho et al.,
2022 Kong al. 2021 Chamberlain et al. (021). Flow matching, in particular, offers conceptu- ally simple and computationally efficient framework, achieving high-fidelity sample generation with fewer function evaluations (Fisser al. 2024

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R(ms)
11
X
++
dt
Physical Residual
CR.
Non-oonfleting
up-iate
Xni
Unrolling
LFM
PBFM
0=
Figure I: Overview of the proposed Physics-Based Flow Matching (PBFM) approach. During training. the sample at time t is evolved to t = 1 over time steps to compute the residual 1().). The flow matching loss LFM and residual loss LR are combined in a conflict-free manner.
Despite recent progress, integrating physical constraints into generative models remains challenging. Enforcing such constraints during training requires the model to jointly optimize for generative fidelity and physical consistency, often resulting in conflicting gradients that can compromise either distributional accuracy or physical validity (Krishnapriyan et al., (022)) Achieving a balanced trade- off between these objectives is difficult, as improvements in one can degrade the other. In this work, we revisit maysscs--cossrrinee generative modeling from a Pareto perspective, explicitly targeting solutions that balance physical correctness and generative performance; systematically analyze the trade-off's involved.
Additionally, the so-called Jensen's 's gap exists: a fundamental discrepancy that arises when physical constraints, such as PDE residuals, are imposed on the posterior mean E [ [ra] at intermediate noise levels, rather than directly the final clean sample 2 (Zhang & Zou 2025). Alternatively, enfore- ing constraints only at inference time typically requires iterative sampling procedures to achieve physical consistency, which can be computationally expensive and undermine the efficiency of gen- erative models. Moreover, enforcing nonlinear hard constraints necessitates differentiating through the physical residual, a process that computationally expensive for high- nigh-dimensional and complex systems (Cheng et al. 2025).
We improve constrained flow matching with series of established techniques such as un- rolling, residual-based losses, stochastic sampling, and conflict-free gradient updates (Liu et al.) 2025a), adapting them to address the unique challenges of physics-constraneed generative modeling. Through the lens of the inherent physics-s-ddistribuie trade-off, we show that these modifications yield very substantial improvements over previous work, and enable training networks that yield distributional accuracy and satisfy physics constraints at the same time. Fig. Fillustslsss the overall training procedure and highlights the key components of PBFM method. Our main contributions follows. (I) We introduce method that integrates physical constraints into flow matching. enabling the minimization of both PDE and algebraic residuals without manual balancing. (II) We show that unrolling during training effectively mitigates Jensen's 's gap, leading lower residual er- rors and improved final predictions without increasing inference cost. (III We analyze the role of Gaussian noise in flow matching under physical constraints, and (IV) we conduct comprehensive comparison between deterministic and stochastic flow matching samplers, demonstrating advan- tages of the latter. central advantage of the proposed approach is that it is very easy implement in existing flow matching pipelines, and, as we will demonstrate below, consistently achieves sub- stantial improvements in terms of distributional and physical accuracy.
2 RELATED WORK
Numerous deep learning methods have been developed to address complex problems in physics and
et al., 2020 Thuerey et al.) 2020). More recently, there has been growing interest in bringing the foundation model paradigm to scientific machine learning. Efforts such as PDEformer (Ye et al. 2004)), Poseidon (Herde et al. 2024). Aurora (Bodnar et al. 2024), and Unisolver (Zhou al. (025a) aim to build models with broad generalization capabilities across diverse physical systems. In parallel, specialized trans- former architectures tailored for PDEs have also been proposed, including OFormer (Li al. 2023
engineering [Morton et al. 2018 Wang et al. 2020 Sanchez-Gonzalle
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Transolver (wu et al. 2024). Fengbo (Pepe al. 2025), and PDE-Transformer (Holzschuh al. 2025).
Focusing difffusion models in the physics and engineering fields, representative applications in- clude the generation and design of new molecules and drugs (Guo et al. 2024 Schneuing et al, 2024 Bose al. 2024 Guastoni & Vinuesa 2023). the simulation of particle trajectories in col- lider experiments (Mikuni et al. 2023), solving inverse PDE problems (Holzschuh et al.) 2023). and the modeling of particle motion turbulent flows (Li al. 2024). Alongside these efforts, several works have sought to embed physical knowledge or constraints directly into the diffusion process enhance model performance. For instance, Huang et al. (2024) presented an approach solve PDEs from partial observations by filling in missing information using generative priors. (Zhou et al. (2025b) proposed an approach that embeds priors into the diffusion process to satisfy energy and momentum conservation laws and PDE constraints. Rixner & Koutsourelakis (2021) formulated probabilistic generative model enforcing physical constraints through virtual observables.
More aligned with our work, Shu et al. 2023) proposed an algorithm that conditions the diffusion process on the residual gradient both at training and inference time, with focus on super-resolution and reconstruction tasks based on random measures of turbulent flows. Despite being one of the first works in this direction, they do not directly enforce the residual. Similarly, JJacobsen et al. 2025) introduced CoCoGen, method that employs the governing PDEs during inference, while leaving the training procedure unchanged. The approach improves the physical residual but slows down inference, which is crucial in real applications. Other approaches that enforce physical straints at inference time include FFM (Kerrigan et al. (023). DiffusionPDE (Huang et al. 2024), 2025), and PCFM (Utkarsh et al. (025). These
D-Flow (Ben-Hamu al. 2024), ECI (Cheng methods require iterative sampling procedures to achieve physical consistency, which can be compu- tationally expensive and can take more than 10x longer than the standard diffusion process (Utkarsh et al.. 2025). The last two methods are designed to strictly satisfy the constraints, but the ECI method limited to simple non-overlapping constraints, while PCFM is applicable arbitrary constraints but requires differentiating through the residual, which is computationally expensive. In contrast, Bastek et al. (2025) proposed physics-informet diffusion models (PIDM), which incorporate an ad- ditional loss term during training to minimize physical residuals. Zhang & Zou (2025) introduced physics-informat distillation diffusion models (PIDDM), to potentially solve the Jensen's 's gap issue by distillation that is fine-tuned to minimize the physical residual. However, this approach requires training two separate models and, like Bastek et al.] (2025) does not resolve the inherent conflict between the generative and physical objectives. Other recent methods that adopt fine-tuning to reduce the physics constraints are PIRF (Yuan et al. (2025) and PCFT (Tauberschmidt et al., 2025j. We provide comparison our method with existing phy cinssscontrrinee generative mod- els in [
Table Comparison between our method and other physics-constraine generative models.
Physics at Hard Training Constraints
Gradient Free Complex Balanced Constraints
Method
Inference
Hyperparams
FFM 34
ProbConsery
24
CoCoGen
32
X
X
X
x
DiffusionPDE 11
X
X
X
PIDM 99
XX
X
XXX
X
D-Flow
ECI [16
X
PCFM (64
PIDDM [71
X
PBEM (ours)
X
METHODOLOGY
PRELIMINARIES
A general time-dependent PDE in spatial dimensions be written uz(x,t) t) Llu(x, t)] + f(x, t), x E Ω C R", where u(x,t) t) is the solution field, L is a spatial differential operator, and f(x, t) denotes external forcing. For numerical approximation, the domain Ω and its boundary
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052 are discretized, and continuous operators replaced by discrete analogues (Karniadakis & Sherwin 2005).
Physics constraints can be enforced by minimizing residual function, which is constructed drawing from wide range of formulations. In generative modeling, the stochasticity of residuals can be categorized into three main types: (i) For steady-state PDEs, the solution distribution arises from uncertainties in the physical parameters, and the residual is computed directly from the governing O. (ii) For time-dependent PDEs, the solution evolves over time and residuals are formulated to enforce conservation laws (such as mass, momentum, or energy) across temporal snapshots. (iii) Algebraic constraints can be used impose additional consistency between physical fields. further enforcing the underlying physics of the system.
equations, e.g., R(C(u) +
3.2 PHYSICS-BANE BASED FLOW MATCHING
Integrating physical constraints into generative models introduces an inherent conflict: optimizing for physical fidelity often degrades distributional accuracy, and vice versa. Existing iiffusion-basee approaches (Shu et al. 2023 Bastek et al. 2025) typically employ a weighted objective:
arg max (ggennnnnnnnnllonon [log pa (21)) + (x1))((z1) [log gr (f 0 [1)]]
(1)
where f are virtual observables (Rixner Koutsourelakis 2021 of the residual R(x1). For flow matching, this reduces to (Bastek et al. 2025j:
C = WPM CFM + wRCR
Frmllug (x,, t) u. (xx))ll + ux [ f))l2.
where WPM and WiR are adjusted manually to address the potential conflicts between generative and physical losses. However, tuning these weights is challenging: increasing the residual term often harms generative quality, while prioritizing the generative term undermines physical consistency.
To address this, leverage multi-task optimization techniques that resolve conflicting gradient directions. Specifically, we adopt the ConFIG method (Liu et al. 2025a) to compute conflict-free updates for physics-based flow matching:
gupdate (BFMgo + ggg.
(3)
go 4 (4(O BR)) + ((O(gR
(4)
where grm and gR are the gradients of LrM and CR. respectively; (gg,, 82) = g2 | gi is the orthogonality operator, and u(g) = g/lg/ normal- izes to unit length. The resulting update direction Supdate guarantees simultancous descent on both ob- jectives: Bupdate > 0 and 8u > 0. Fig-
ki
Pupdate
9FM
ure represents the gradient composition.
This approach adaptively aligns the gradients, elim- inating the need for manual loss weighting and pre- venting one objective from dominating optimiza- tion. As a result, the resulting updates yield high distributional accuracy while still enforcing phys- ical constraints, consistently outperforming fixed- weighted objectives across wide range of weights (see Appendix [ for an analysis).
9R
Funrolling
Figure 2: PBFM method leverages ConFIG to combine the unit vectors of the orthogo- nal components and scales the result using the projection length of the FM and the physical residual R gradients.
3.3 IMPROVING PHYSICAL AND DISTRIBUTIONAL ACCURACY
A key requirement for generative models scientific applications the ability to produce samples with high physical accuracy, close to traditional numerical methods. In physics-constrained genera- tive modeling, a central challenge is the so-called Jensen's gap which appears whenever a nonlinear map f is applied to random variable Z: in general [[f( # f(B[Z)) When physical con- straints are imposed on the posterior mean E[2 1 [x,] at intermediate noise levels, rather than directly on the final clean sample æ1, discrepancy arises that can degrade physical fidelity (Zhang & Zou
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2023). To mitigate this, we identify the reconstruction of the final, frese--fre sample during training as crucial for accurately evaluating the physical residual. Specifically, we employ unrolling, i.c., integrating the intermediate sample at time t forward to the final time t = 1 using multiple ODE steps. This process mitigates Jensen's 's gap by ensuring that the residual is evaluated on a more ac- curate prediction of 31; rather than single-step approximation. By unrolling over steps of size (1-t)/n, the integration better approximates the true trajectory. Unrolling is applied via a curricu- lum, gradually increasing the number of steps during training. Additionally, we down-weight less accurate predictions near t = 0 by applying a scaling factor t" (with Pogt = 1) to the residual loss, further improving the learning signal. Linear scaling is also principled, aligning with the linear noise schedule in flow matching and ensuring consistency between residual weighting and FM dynamics. An investigation of different power laws is provided in Appendix F The improved prediction of z substantially enhances the evaluation of the physics residual in Eq. 2 yielding better optimization directions and directly mitigating Jensen's 's gap. This leads to lower residual errors and improved final predictions, without increasing inference cost. Nevertheless, unrolling increases memory con- sumption during training, as intermediate states must be stored for backpropagation. Table 6 details the additional training time and memory requirements for different numbers of unrolling steps.
Another important aspect in physics- based flow matching is the choice of minn the amount of Gaus- sian noise added to training data adhering to the mixture of Gaussians theory. While computer vision tasks typically use Amin = 10-3 (Lipman et al. 2023 Esser et al. 2024), excessive noise per- turbs physical residuals and degrades performance. The value of m thus influences the minimum achievable residual error. We provide an analysis of different noise levels in Table 5 A practical guideline that adding Gaussian noise of scale min induces residual MSE m perfect reconstruction setting, reducing to m S 'Rmin-
Inspired by natural image generation (Esser et al.) 2024), we also sample the time variable from a logit-normal distribution (zero mean, unit standard deviation) during training, instead of a uniform distribution. This specifically targets regions where flow matching exhibits higher errors, typically around / 0.5.
Algorithm details the resulting training procedure, where the FM loss is computed at time t and unrolling is used for accurate residual evaluation. The initial time is stored to weight the residual loss.
Algorithm Training procedure for PBFM
n < number of unrolling steps
dt < (1 t)/n [+1
Compute di reach the final state t >. Keep the starting time to weight the residual loss
" + model( t)
> Flow matching velocity uf is not part of unrolling
31 Ia xi+ dt uf
for i = =1, -1,i< n do
> Improved state prediction via unrolling
i= f=t+d dt
ue + model i)
31 + Fn + dt 7
> Current fields F1 are updated until t 1
end for
R + compute residual (z1)
LR - ||00 R1
> Residuals are weighted with (2, Popt
LFM + luf uzlla
Vo < compute Supdate via Eq. 3
> Conffict-free update
AdamW optimizer step with
At inference time, the final sample typically obtained by evolving the initial noise, (0, I) with ODE integration. This represents deterministic sampler in which all the stochasticity is embedded in the initial noise sample (Gao et al. 2025). In contrast, the sampling process in DDPM is stochas- tic (Ho et al.) (202)). Given that both diffusion models and flow matching can be seen generative modeling variants under arbitrary Markov processes (Holderrieth et al. 2025). we explore the use of stochastic sampler, similarly to what has been applied in ECI method (Cheng et al. 2025). within the physically based flow matching framework. The central idea is to evolve from time to = 1 and then return to + dt using a different noise sample. This step backwards in time with newly gener- ated noise increases the stochasticity in the sampling process and improves distributional accuracy. The resulting procedure is outlined in A Algorithm
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4 EXPERIMENTAL SETUP
We evaluate the generative performance of our method, denoted with Physics Based Flow Match- ing (PBFM) in the following, on three benchmark problems. Across all experiments, we employ diffusion transformer (DiT) backbone architecture (Peebles & Xie 2023), with minor modifica- tions detailed in Appendix G For completeness, we also provide a comparison to the UNet from previous work (Ronneberger method is agnostic to the type of physical residual, we organize the three benchmarks according to the residual categories described above: steady-state, transient, and analytic. For each case, the residual is computed using: finite differences for Darey flow, FFT-based methods for Kolmogorov flow, and point-wise evaluation for dynamic stall. complete description of datasets is provided in Appendix
al.) 2015 Bastek et al.) 2025), shown in Fig. [0] Although our
Darcy flow We begin with the two-dimensional Darey flow problem. The Darey equation which models steady-state fluid flow through a porous medium. The solution comprises pressure p and per- meability K. We use the corresponding public dataset (Bastek) 2024), which contains 10k training and Ik validation samples, each of size 64 x 64. It is worth noting that, while common litera- ture (Zhu & Zabaras 2018 Jacobsen et al. 2025), this dataset lacks conditioning inputs, making less representative real real-world application scenarios. The residual directly corresponds to the governing PDE: R V (KVp) + ==
Kolmogorov flow The second benchmark is the two-dimensional Kolmogorov flow over 128 x 128 spatial domain with periodic boundary conditions. The dataset is generated using a spectral solver and consists of velocity field snapshots for various Reynolds numbers the range [100; 500]. The Reynolds number conditions the data generation, influencing the turbulence scales and flow complexity. The training set includes data for 32 Reynolds numbers with 1024 temporal snapshots each. The test set additionally includes 16 unseen Reynolds numbers. The data distribution reflects the temporal variation within each flow regime. The prediction target consists of the two velocity components, which are expected to satisfy the conservation of mass via: R V U
Dynamic Stall The final and most complex setup involves spatio-temporal fields over a pitch- ing NACA0012 airfoil. This case captures the effects of dynamic stall, a complex, unsteady phe- nomenon in aerodynamics causing large-scale fluctuations and loads. As such, this physical model is highly relevant for real world cases such as helicopter and wind turbine blades, where dynamic stall plays a critical role. The solutions and the flow fields present shock waves and detailed flow phenomena that are challenging to capture. Each sample consists of 128 128 distributions of six quantities: absolute pressure, temperature, density, skin friction, and tangential velocity gradients (x and y). Conditioning is performed on four parameters that define the operating conditions and airfoil motion. The training set comprises 128 base configurations, each perturbed 32 times to model uncertainty. while the validation set includes 16 unseen configurations. Physical consistency across fields is enforced through two analytical, point-wise residual constraints: Rig imposes the ideal gas law, while R, minimizes the skin friction with Sutherland's law.
obtained solving the compressible Navier-Stokes equations
To S
T 3
Ou Fx
2
Rig = P - -RRT',
R₂ = =T 140
T
S
To
1
+
Oy
(5)
RESULTS
We evaluate our proposed framework, PBFM (Baldan et al. 2025a). in direct comparison with six representative baselines: flow matching with optimal transport FM-OT trained without phys- ical constraints, PIDM-ME (Bastek et al.) 2025), CoCoGen (Jacobsen et al.) 2025), Diffusion- PDE (Huang et al. 2024), D-Flow Hamu et al.) 2024). and ECI (Cheng et al. 2025). ECI only enforces BCs but not the PDE due to the limitations of the method. For each method, report physical residual error, distributional fidelity via Wasserstein distance and Jensen- Shannon divergence, number of function evaluations, and inference wall-clock time per sam- ple. For the conditional benchmarks, Kolmogorov flow and dynamic stall, we additionally eval-

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Table 2: Generative performance metrics for Darcy flow problem 1024 samples. RE: physi- cal Residual MSE, WD: Wassserstein Distance, JS: Jensen-Shannon divergence, NFE: Number of function evaluations, IT: Inference wall-clock time. Best and second best results.
Metric PBFM FM-OT CoCoGent PIDM
DiflusionPDE D-Flow
ECI$
RE
0.838
4.159
1.320
0.022
3.388
2.286
3.045
WD -102 0.138
0.059
0.249
3.103
0.089
0.147
2.892
IS -00
0.256
0.131
0.360
3.179
0.139
0.237
2.818
NFE
20
20
100
100
20
20
20
IT
0.101
0.100
7.395
2.050
0.590
3.126
0.122
1
This method uses the UNet architecture from Bastek et al.
2025
$ D-Flow method unstable, samples are filtered using RE
5 condition resulting in 888 valid ones.
$ The physical constraint is applied only to the BCs (REac 0) but cannot be applicd to the non-linear PDE.
uate the mean and standard deviation of the predicted distributions for each conditioning in- put. Comprehensive sample visualizations and further analysis are provided in Appendix
PIDM-ME
ECT.
5.1 DARCY FLOW
Wasserstein Distance
Inference Time [
10-2.
Unlike previous works that report only individ- ual metrics, we present comprehensive eval- uation of Pareto optimality in terms of phys- ical residual error and distributional accuracy.
CoCoGen
The latter is quantified by the Wasserstein dis-
PBFM (Ouns)
D. Flow
10-3
tance (WD) (Villani 2008 SSiPyy 2025). Fig. 3 visualizes the trade-off for all compared meth- ods. The FM-OT baseline yields WD of 0.059 but incurs a high residual error of 4.159. Diffu- sionPDE achieves similar WD with marginally improved residuals. CoCoGen and D-Flow are strictly dominated by our approach, exhibiting both higher residuals and inferior distributional
FM-OT DiffussonPDE
10-1
10
10L
Plysical Residual MAE
Figure 3: Pareto front of physical residual MAE vs. Wasserstein distance for Darcy flow for the proposed methods.
metries. PIDM-ME attains the lowest residual (0.022), which comes at the expense of a substantially degraded distribution (WD 3. 3.103), reflecting poor generative fidelity. In contrast, PBFM achieves residual of 0.838 and WD 0.138, demonstrating more favorable balance between physical and generative objectives. A visual representation of the residual errors is reported in Fig. 5 The pre- vious methods align along a negative-slope trend, highlighting the intrinsic trade-off between resid- ual minimization and distributional accuracy. PBFM advances the Pareto front, reducing residuals without a considerable increase in WD. Comprehensive metrics are reported Table 2 including Jensen-Shannon divergence (JS) (Lin, 199T> as distributional metric in addition WD, the num- ber of function evaluations (NFE), and inference time (IT). PBFM maintains competitive inference speed (0.101 s), similarly to FM-OT, as the only difference is the use of the stochastic sampler.
The first panel of Fig. illustrates two aspects. On the hand, shows how residual enror evolves with the number of unrolling steps used during training and it demon- strates that unrolling effectively miti- gates Jensen's 's gap, leading to lower residual errors. On the other hand, it highlights that only applying the ConFIG method during training of- fers a limited improvement. To fur- ther analyze the impact unrolling, the second panel of Fig. presents
urolinn anciling unnding undling dirp
10
10
staps
stape
stape
Ryos
10%
109
0.5
1.0
20
50
Training Stepe. X10
FM steps
FM-OT
FM-OT + Cg with Config
PBFM
Figure 4: Darcy flow validation over 1024 samples using 20 FM steps. Residual MAB as function of training steps (left), residual MAE as a function of FM steps (right).
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the residual error a function of the number of FM steps used during inference. The results confirm that increasing the number of FM steps consistently reduces the residual error across all methods. Notably, PBFM delivers the lowest residual enror and it is less sensitive the number of FM steps, indicating that achieves high physical accuracy even with fewer function evaluations. This is particularly advantageous in practical applications where computational resources are limited.
To complete the analysis, we evaluate the impact of the stochastic sampler introduced in Algorithm 2 Table 3 reports the performance of PBFM with different thresholds for noise resampling during inference. Setting t" 0.0 corresponds to the deterministic sampler, while 1.0 implies resampling noise at every step. The results indicate that using the determin- istic sampler yields comparable residual emoor the stochastic approach, but with significantly worse distributional metrics. Focusing on values greater than O, we observe that increasing 1 leads improved residual error, with the best performance at t* 1.0. However, this comes at the cost of degraded distributional accuracy, as both WD and JS increase. A balanced choice of t = 0.2 offers a good compromise, achieving low residuals while maintaining strong generative fidelity. Further results showing pressure and permeability samples and an additional UNet are reported in
Table 3: Performance of stochastic sampler for Darey flow prob- lem over 1024 samples using 20 FM steps using PBFM. Noise resampling is enabled for < t*, details in Algorithm 2 t 0.0 coincides with deterministic sampler. Best and second best results.
t"
Metric
0.0
0.2
0.4
0.6
0.8
1.0
RE WD -102
0.828
0.838
0.970
0.876
0.774
0.632
1.470
0.138
0.150
0.187
0.302
0.316
JS -10
0.919
0.256
0.257
0.313
0.361
0.379
Fig. 9 and of the appendix.
FM-OT
PBFM (Ours)
10
5.2 KOLMOGOROV FLOW
RMAE(K p))
10t
Kolmogorov flow presents a conditional gener- ative modeling challenge, where the Reynolds number serves as the conditioning input and the physical residual is defined by the divergence of the velocity field, enforcing conservation of mass. Table 4 details the quantitative results. PBFM achieves the lowest physical residual er- ror (1.362), outperforming all baselines. It also yields the best distributional metrics, with a low Wasserstein distance and
10000
10-1
Figure 5: Physical residual of Darcy flow exam- ples for the proposed method with 20 FM steps.
divergence, demonstrating excellent generative fidelity. The standard deviation error slightly higher than FM-OT, but the overall distributional accuracy remains superior. Inference times for PBFM (98.97 ms) are competitive with FM-OT, while Dif- fusionPDE and D-Flow incur substantially higher computational costs due to FFT-based residual evaluation. D-Flow is unstable for Kolmogorov flow and does not produce valid samples despite the high computational cost.
Qualitative results provided in Appendix Fig. 12 and 13 which show representative instanta- neous velocity fields, divergence residuals, and the predicted mean and standard deviation across the test set.
Table 5: Metric comparison for different values of Omin on the dynamic stall problem, 20 FM steps. Best and second best results.
5.3 DYNAMIC STALL
The dynamic stall case presents additional chal- lenges, including two physical constraints of increased complexity and six predicted fields. The underlying phenomena are highly non- linear containing shock waves. Fig. 66) presents an example of predicted fields. Especially re- gions like the center exhibit strongly varying,
Metric
T min
0.0
10
10
10-2
RE 10
0.339
0.468
0.473
0.466
WD -10
1.814
3.547
3.246
3.566
JS -102
0.680
0.716
0.718
0.728
MMSE -10
1,490
1.360
1.298
1.200
small-scale shock waves that stochastically os- cillate for perturbed operating conditions phenomenon that is important to capture for industrial applications (Baldan al.
SMSE -10
0.874
0.831
0.789
0.916
2025b).
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Table 4: Generative performance metrics for Kolmogorov flow and dynamic stall problems using 20 FM steps. RE: physical Residual MSE, WD: Wasserstein Distance, JS: Massen-Shannon divergence, MMSE: MSE of the mean fields, SMSE: MSE of the standard deviation fields, NFE: Number of function evaluations, IT: Inference wall-clock time. Best and second best results. Dataset
Metric
PBFM OT-FM
DiffusionPDE D-Flow
PCEM
RE 10-
1.362
2.314
1.930
WD -101
1.222
2.124
3.698
Kolmogorov
JS -102
7.440
12.49
19.39
Flow
MMSE -104
4.455
4.188
5.669
SMSE -144
2.484
2.574
21.82
IT mms]
98.97
98.75
267.8
6431
RE -10
0.339
11.02
12.20
11.32
0.143
WD -104
1.814
2.707
2.509
3.484
4.013
Dynamic Stall
IS -102
0.680
0.983
1.029
1.014
1.206
MMSE -105
1.490
2.791
2.626
2.507
5.669
SMSE -105
0.874
1.458
1.236
1.372
7.674
IT [ms]
60.47
59.75
171.7
138.9
3906
$ D-Flow method is unstable Kolmogorov Flow.
Table 4 summarizes the quantitative results. PBFM outperforms all baselines across nearly all met- rics, achieving the lowest distributional errors and MSE for both the mean and standard-deviation fields, which indicates that PBFM effectively captures both first- and second-order statistics of the complex flow fields. PCFM achieves the lowest residual error, but this comes at the cost of worse distributional metries and higher mean and Htandard-deviatee nificantly higher inference time (> 65x that of PBFM) due to the iterative correction process. Inference times for PBFM are comparable OT. DiffusionPDE and D-Flow incur higher compu- tational costs because evaluating their phy sical resid- uals is more complex, although the cost difference is limited since the residuals are pointwise.
errors. Notably, PCFM incurs a sig-
FM-
Mean P
Mean Tu
Reference
To investigate the impact of different values of miin
on the dynamic stall problem, which present the lowest residual errors and is more sensible this phenomenon, Table [ analyzes the results. Setting Tmin yields the best distributional metrics and residual error underlines the importance of minimiz- ing noise perturbation in this complex scenario. In- creasing min to 10- slightly degrades the met- rics, while further increases to 10 and 10 lead to more pronounced declines in performance. This trend highlights the sensitivity of physical residu- als to noise levels, emphasizing the need for care- ful tuning of Jmin in physics-constrained generative modeling. Nevertheless, with higher noise levels the model provides lower mean and standard deviation errors. A practical guideline is that adding Gaussian noise of scale amin induces a residual MSE ~ o in perfect reconstruction setting. For dynamic stall, this directly suggests upper bound: 22 < 10-7 + min S 3 x 10-4, In contrast, Kolmogorov and Darcy benchmarks have more relaxed residual requirements. Importantly, this sensitivity is not unique to our method; it is general to flow matching formulations, as the noise level determines the scale of the denoising target and hence the attainable physical accuracy.
50000
100000
-50
0
250
PBPM (Onrs)
50000
100000
250
250
Figure 6: Dynamic stall, comparison of mean for PBFM framework computed over 128 samples using 20 FM steps.
Performance evaluation We provide an overview of the training performance, focusing on the impact of the residual loss together with ConFIG, and the effect of unrolling. Table 6 reports the

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Published as a conference paper at ICLR 2026
wall-clock time, in seconds, for single training iteration NVIDIA A100 64 GB GPU. To reduce training computation time and memory footprint, backpropagation can be computed only through the final unrolling step; this does not significantly affect model performance. In Appendix we provide a detailed comparison of this variant against full backpropagation through all unrolling steps.
Table 6: Comparison of wall-clock time in seconds for one training iteration and memory usage in GB on an NVIDIA A100 64GB GPU for the proposed approaches. Batch size is 64 for all cases. Inference time is unchanged.
FM
PBFM step
PBEM steps
PBFM steps
PBFM steps
Darey
[s]
4.28 - 10
6.69 10
10-2 23.26
9.78
1.28 10-1 33.70
10 44.15
1.59
[GB]
12.35
12.62
Kolmogorov
[s]
1.13 10-1
1.94 10-1
3.02 10-1
4.10 10-1
5.18
10
[GB]
4.08
4.52
6.91
9.56
12.18
Dynamic
[s]
4.29
10
8.14
10-2
1.18
10-1
1.55
10-1
1
[GB]
4.25
4.41
7.18
9.90
12.58
DISCUSSION AND CONCLUSIONS
Limitations PBFM delivers substantial improvements in physical and distributional accuracy, but its main limitation is increased computational and memory cost during training. Table (6) details the wall-clock time and memory usage for different unrolling steps. Incorporating the residual loss and ConFIG requires an additional backward pass, increasing training time and memory usage. Unrolling over 4 steps further increases memory consumption (up to 3x) and training time (up to 2.5x). However, these overheads are restricted to training; the inference speed of the method remains unaffected.
Discussion PBFM designed enable rapid prediction of complex physical systems, particularly in scenarios where conventional solvers are prohibitively slow. Table To presents inference times for the dynamic stall benchmark, already employed for actual helicopter blade simulations. While single simulation using CPU-based solver requires approximately 76 minutes, the trained PBFM model generates 128 samples in just minutes on CPU and only 0.2 seconds on GPU. This substan- tial acceleration demonstrates the practical utility of PBFM for large-scale uncertainty quantification and surrogate modeling tasks in scientific and engineering workflows. Without the proposed modi- fications, trained model would not be sufficiently accurate for real-world applications.
Summary In this paper, introduced PBFM, generative framework designed to improve phys- ical consistency while preserving the strengths of the flow matching approach for high-dimensional data generation. Our model provides straightforward improvements mize physical residuals, arising from PDES or algebraic constraints, in a conflict-free manner. We conduct extensive benchmarks across three representative physical systems, demonstrating the ver- satility and robustness of our approach. The incorporation of temporal unrolling plays key role, enabling improved final state approximations and mitigating Jensen's ' gap by providing more ac- curate prediction of the final sample. Additionally, our results highlight the benefits of stochastic sampling strategics, which outperform deterministic methods in cases involving complex target dis- tributions. Beyond these specific benchmarks, the proposed framework generalizes to wide class of constrantdd problems formulated via residual minimization. By combining the computational efficiency of flow matching with the interpretability and rigor of physics- based modeling, PBFM of- fers a powerful and flexible tool for surrogate modeling, uncertainty quantification, and simulation acceleration in physics and engineering applications.
standard FM to mini-
10

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Published as a conference paper at ICLR 2026
REFERENCES
Moloud Abdar, Farhad Pourpanah, Sadiq Hussain, Dana Rezazadegan, Li Liu, Mohammad Ghavamzadeh, Paul Fieguth, Xiaochun Cao, Abbas Khosravi, U. Rajendra Acharya, Vladimir Makarenkov, and Saeid Nahavandi. A review of uncertainty quantification in deep learning: Techniques, applications and challenges. Information Fusion, 76:243-297, 2021. ISSN 1566- 2535. doi: ht0.1//6.inffus.222105555555
URL https://wwwwww. http::/:wwww..cieeeeeeeeeeee.e com/
ANSYS. Fluent 2024R2, 2024. URL
wwwwwansyy.oo www.waanss..com/ppooouccs////uuu
ansys- fluent
Giacomo Baldan and Alberto Guardone. A deep neural network reduced order model for unsteady aerodynamics of pitching airfoils. Aerospace Science and Technology. 152:109345, 2024. ISSN 1270-9638. doi:
109344. URL htt.h: www. www.sciieeneeeee www.scciencedreet....... com/
Giacomo Baldan and Alberto Guardone. Wall-resolved large eddy simulations of pitching airfoil in deep dynamic stall. Physics of Fluids, 37(2):02412,, 02 2025. ISSN 1070-6631, doi: 10.1063/ 5.0252828.
Giacomo Baldan, Qiang Liu, Alberto Guardone, and Nils Thuerey. Flow matching meets PDEs: unified framework for physics-constrained generation, 2025a. URL https::/arxiv..ot
abs/25055 08604
Giacomo Baldan, Francesco Manara, Gregorio Frassoldati, and Alberto Guardone. The effects of turbulence modeling on dynamic stall. Acta Mechanica, 236(2):1411-1227, 2025b. ISSN 1619- 6937. doi: 10.1007/500707-025--4222
Marco Baldan, Giacomo Baldan, and Bernard Nacke. Solving 1D non-linear magneto quasi-static Maxwell's equations using neural networks. IET Science, Measurement & Technology, 15(2): 204-217, 2021. doi: 10. 10.1049/smt2. 12022.
Jan-Hendrik Bastek. Physics-Informed Diffusion Models. Datasets and model checkpoints CC BY 4.0, 2024.
Jan-Hkk Bastek, WaiChing Sun, and Dennis Kochmann. Physics-Informed Diffusion Models. In The Thirteenth International Conference on Learning Representations, 2025. URL https:
pweeneeveew.ntt/oornnne
Heli Ben-Hamu, Omri Puny, Itai Gat, Brian Karrer, Uriel Singer, and Yaron Lipman. D-Flow: differentiating through flows for controlled generation. In Proceedings of the 41st International Conference on Machine Learning, ICML'24. JMLR.org. 2024.
Cristian Bodnar, Wessel P Bruinsma, Ana Lucic, Megan Stanley, Johannes Brandstetter, Patrick Garvan, Maik Riechert, Jonathan Weyn, Haiyu Dong, Anna Vaughan, al. Aurora: foundation model of the atmosphere. arXiv preprint arXiv:2405. 13063, 2024.
Joey Bose, Tara Akhound-Sadegh, Guillaume Huguet, Kilian FATRAS, Jarrid Rector-Brooks, Cheng-Hao Liu, Andrei Cristian Nica, Maksym Korablyov, Michael M. Bronstein, and Alexander Tong. SE(3)-stochastic flow matching for protein backbone generation. The Twelfth Interna- tional Conference Learning Representations, 2024. URL https:://penrreieew
Forum?id=kJF 1H2333FJJFH333
Steven L. Brunton and J. Nathan Kutz. Promising directions of machine learning for partial differ- ential equations. Nature Computational Science, 4(7):483-494 Jul 2024. ISSN 2662-8457. doi:
Ben Chamberlain, James Rowbottom, Maria I Gorinova, Michael Bronstein, Stefan Webb, and Emanuele Rossi. GRAND: Graph Neural Diffusion. In Proceedings of the 38th International Conference on Machine Learning, volume 139 of Proceedings of Machine Learning Research, pp. 1407-1418. PMLR, 18-24 Jul 2021.

---

Published as a conference paper at ICLR 2026
Wenqian Chen, Qian Wang, Jan S. Hesthaven, and Chuhua Zhang. Physics- machine learn- ing for reduced-order modeling of nonlinear problems. Journal of Computational Physics, 446: 110666, 2021. ISSN 0021-9992 doi: 10.00166..ep.2221111666
Chaoran Cheng, Boran Han, Danielle C. Maddix, Abdul Fatir Ansari, Andrew Stuart, Michael Mahoney, and Bernie Wang. Gradient-free generation for hard-constrained systems. In The
Thirteenth International Conference on Learning Representations, 2025. URL https://
openreview. http:iiwwwewttteemmm wtw.:/orrmmm//o wtw.ooormmmnaaa11111ee00
Patrick Esser, Sumith Kulal, Andreas Blattmann, Rahim Entezari, Jonas Müller, Harry Saini, Yam Levi, Dominik Lorenz, Axel Sauer, Frederic Boesel, Dustin Podell, Tim Dockhorn, Zion En- glish, Kyle Lacey, Alex Goodwin, Yannik Marck, and Robin Rombach. Scaling Rectified Flow Transformers for High- Resolutes Image Synthesis, 2024. wttpthtss////rrii wrg. abs/ 03206
Lawrence C. Evans. Partial differential equations. American Mathematical Society, Providence, R.I., 2010. ISBN 9780821849744 0821849743.
Stefania Fresca, Luca Dede", and Andrea Manzoni. A Comprehensive Deep Learning- Based Ap- proach to Reduced Order Modeling of Nonlinear Time- teme-Dependent Parametrized PDEs. Journal of Scientific Computing, 87(2);61, Apr 2021. ISSN 1573-7691. doi: 10. 10.110777110915--1--01463
Ruiqi Gao, Emiel Hoogeboom, Jonathan Heek, Valentin De Bortoli, Kevin Patrick Murphy, and Tim Salimans. Diffusion models and gaussian flow matching: Two sides of the same coin. In
The Fourth Blogpost Track at ICLR 2025, 2025. URUJhttps://ppeneviiee wwwpiiewwwwe/..o.oeeoeeeeeooo formm??
Id=C8Yyg9wy/U
Luca Guastoni and Ricardo Vinuesa. A new perspective on the simulation of stochastic problems in fluid mechanics with diffusion models: Fluid dynamics. Nature Machine Intelligence, pp. 1-2, 2025. doi: 10. 10..11///2255-055--00000
Zhiye Guo, Jian Liu, Yanli Wang, Mengrui Chen, Duolin Wang, Dong Xu, and Jianlin Cheng. Diffusion models in bioinformatics and computational biology. Nature Reviews Bioengineering, 2(2):136-154, Feb 2024. ISSN 2731-6092. doi: 101033/54222-22-001144.
Eldad Haber, Lars Ruthotto, Elliot Holtham, and Seong-Hwar Jun. Learning across scales multiscale methods for convolution neural networks. In Proceedings of the AAAI conference on artificial intelligence, volume 32, 2018.
Derek Hansen, Danielle C. Maddix, Shima Alizadeh, Gaurav Gupta, and Michael W. Mahoney. Learning physical models that can respect conservation laws. Proceedings of the 40th Interna- tional Conference on Machine Learning, ICML'23, JMLR.org, 2023.
Maximilian Herde, Bogdan Raonic, Tobias Rohner, Roger Käppeli, Roberto Molinaro, Emmanuel de Bezenac, and Siddhartha Mishra. Poseidon: Efficient Foundation Models for PDEs. In The Thirty-eighth Annual Conference on Neural Information Processing Systems, 2024. URL
https: http:::///ppeeeeeee....m wwtp:/wwwwwwwe..eeeeeem.
Jonathan Ho, Ajay Jain, and Pieter Abbeel. Denoising Diffusion Probabilistic Models, 2020. URL
https: http///axvv... http:///axxvx/.//a./.ao pttp:/awww..g///////......... 11239
Jonathan Ho, Tim Salimans, Alexey Gritsenko, William Chan, Mohammad Norouzi, and David Fleet. Video diffusion models. In Advances in Neural Information Processing Systems, vol- ume 35, pp. 8633-8646. Curran Associates, Inc., 2022.
Peter Holderrieth, Marton Havasi, Jason Yim, Neta Shaul, Itai Gat, Tommi Jaakkola, Brian Karrer, Ricky T. Q. Chen, and Yaron Lipman. Generator Matching: Generative modeling with arbitrary Markov processes. In The Thirteenth International Conference on Learning Represenaations, 2025. URL htll:ttts////peeviiii
net/ feeet/forumiid-220 Ruf Ru
Benjamin Holzschuh, Simona Vegetti, and Nils Thuerey. Solving inverse physics problems with score matching. Advances in Neural Information Processing Systems (NeurIPS), 36, 2023.
12

---

Published as a conference paper at ICLR 2026
Benjamin Holzschuh, Qiang Liu, Georg Kohl, and Nils Thuerey. Tranfoanfform::: Efficient and Versatile Transformers for Physics Simulations. Forty-second International Conference on Machine Learning. 2025.
Jiahe Huang, Guandao Yang, Zichen Wang, and Jeong Joon Park. DiffusionPDE:: Generative PDE- solving under partial observation. In The Thirty-eighth Annual Conference on Neural Information Processing Systems, 2024. URL https: / http:://ppppeeeeeeee wtt.///oootndd-do//aaooaoooooooo
Christian Jacobsen, Yilin Zhuang, and Karthik Duraisamy. CoCoGen: Physically Consistent and Conditioned Score-Based Generative Models for Forward and Inverse Problems. SIAM Journal on Scientific Computing, ((():C339-CC42) 2025. doi: 10.1 10.1137/24M1636077
George Karniadakis and Spencer Sherwin. Spectral/hp Element Methods for Computational Fluid Dynamics. Oxford University Press, 06 2005. ISBN 9780198528692.
doi: 10. 10.1093/acprof:
0so/9780198528892.000 1.0011.1........0000001110000...
Gavin Kerrigan, Giosue Migliorini, and Padhraic Smyth. Functional Flow Matching, 2023. URL
httpps:/aaran org/ abs awwp//////s.. 17209
Ladislav Kocis and William J. Whiten. Computational investigations of low-discrepancy sequences.
ACM Trans. Math. Softw., 23(2):266-294 294, jun 1997. ISSN 0098-3500. doi: 10. 10.1145/264029 264064.
Zhifeng Kong, Wei Ping, Jiaji Huang, Kexin Zhao, and Bryan Catanzaro. Diffwave: A versatile diffusion model for audio synthesis. In International Conference on Learning Representations, 2021. /RRl:tttss:/opeeeevieee http//pperrvveeeeeee net. formm?rrrmmmm d=na-miiiia--- xF
Aditi S. Krishnapriyan, Amir Gholami, Shandian Zhe, Robert M. Kirby, and Michael W. Mahoney. Characterizing possible failure modes in physics-informed neural networks, 2021. URL https:
//arxiv. wwwxiiv.or//aaa barrg//bs//le abb// 01050
T. Li, L. Biferale, F. Bonaccorso, M. A. Scarpolini, and M. Buzzicotti. Synthetic lagrangian tur- bulence by generative diffusion models. Nature Machine Intelligence, 6(4):393 Apr 2024.
ISSN 2522-5839. doi: 10. 00.10888/42226--02-0088888
Zijie Li, Kazem Meidani, and Amir Barati Farimani. Transformer for partial differential equations' operator learning. Transactions Machine Learning Research, 2023. ISSN 2835-8856. URI.
aattps://ppeneeviee net/ forum?id EPPqt.3ERR
J. Lin. Divergence measures based on the Shannon entropy. IEEE Transactions on Information Theory, 37(1):145-151 151, 1991. doi: 10. 105/
Yaron Lipman, Ricky T. Q. Chen, Heli Ben-Hamu, Maximilian Nickel, and Matthew Le. Flow Matching for Generative Modeling. In The Eleventh International Conference on Learning Rep- resentations, 2023.
net http://wwwwww..tteeoooooooo//oo/
Yaron Lipman. Marton Havasi, Peter Holderricth, Neta Shaul, Matt Le, Brian Karrer, Ricky T. Q. Chen. David Lazez-Paz, Heli Ben- and Itai Gat. Flow matching guide and code, 2024.
hbtpg//bb///441// 06264
Qiang Liu and Nils Thuerey. Uncertainty- Surrogate Models for Airfoil Flow Simulations with Denoising Diffusion Probabilistic Models. AIAA Journul, 62(8): 2024. doi:
10.2514/1.J63440..
Qiang Liu, Mengyu Chu, and Nils Thuerey. ConFIG: Towards Conflict-free Training of Physics Informed Neural Networks. In The Thirteenth International Conference on Learning Represen- tations, 2025a. Utt.attpp.//ol...r//00/.2////2/2
/200 11104
Qiang Liu, Felix Köhler, and Nils Thuerey. TorchRSM: Fourier Spectral Method with PyTorch, May 1535020
Vinicius Mikuni, Benjamin Nachman, and Mariel Pettee. Fast point cloud generation with dif- fusion models in high energy physics. Phys. Rev D, 108:036025, Aug 2023. doi: 10.1 SitssReDD110.03666 URL [attrsss//axx...ae..cc..o//./... 10e. 0.36025
13

---

Published as a conference paper at ICLR 2026
Jeremy Morton, Antony Jameson, Mykel J Kochenderfer, and Freddie Witherden. Deep dynamical modeling and control of unsteady fluid flows. In Advances in Neural Information Processing Systems, 2018.
Alexander Quinn Nichol and Prafulla Dhariwal. Improved Denoising Diffusion Probabilistic Mod- els. In Proceedings of the 38th International Conference on Machine Learning, volume 139 of Proceedings of Machine Learning Research, pp. 8162-8171. PMLR, 2021. URL https:
/ htt.:////nnchccch...m.mm html
William Peebles and Saining Xie. Scalable diffusion models with transformers. In 2023 IEEE/CVF International Conference on Computer Vision (ICCV), pp. 4172-411 4122, 2023. doi: 10.1109/
ICCV51000.2023.00D 00387.
Alberto Pepe, Mattia Montanari, and Joan Lasenby. Fengbo: Clifford Neural Operator pipeline for 3D PDEs Computational Fluid Dynamics. The Thirteenth International Confer- Learning Representations, 2025. URL VsxbWTDH
wwwpeenrevvew..eeeeeeeeeee net. wwwp:/rumffformoecommomm
M. Raissi, P. Perdikaris, and G.E. Karniadakis. Physics-informed neural networks: A deep learn- ing framework for solving forward and inverse problems involving nonlinear partial differen- tial equations. Journal of Computational Physics, 378:686-707, 2019. ISSN 0021-9991. doi:
10.1016/j.jcp.211 10.045.
Maximilian Rixner and Phaedon-Stelios Koutsourelakis. A probabilistic generative model for semi- supervised training of coarse-grained surrogates and enforcing physical constraints through vir- tual observables. Journal of Computational Physics, 434: 10218, 2021. ISSN 0021-9991. doi: jo..006/j..cp.2221111.. article/p11/50022 000000021999121000110
URL httpt:tt///// //ws:://www..
contttccoo/e sts//ciienee science/
Robin Rombach, Andreas Blattmann, Dominík Lorenz, Patrick Esser, and Björn Ommer. High- resolution image synthesis with latent diffusion models. In Proceedings of the IEEE/CVF Con- ference on Computer Vision and Pattern Recognition (CVPR), pp. 10684-10695, June 2022.
Olaf Ronneberger, Philipp Fischer, and Thomas Brox. U-Net: Convolutional Networks for Biomed- ical Image Segmentation. In Medical Image Computing and Computer-Assisted Intervention MICCAI 2015, pp. 234-241. Springer International Publishing, 2015.
Christopher J. Roy and William L. Oberkampf. A comprehensive framework for verification, val- idation, and uncertainty quantification in scientific computing. Computer Methods in Applied Mechanics and Engineering. 200(25):2131-244 2011. ISSN 0045-7825 doi: 10.1016/j 2011.03.016. URL https: wwwp www.ncedieicccc..co.oo..e..ec.ce rect. wtt.ee/cco//ceeeeeeaaaiiileeee/e 5004518251100
Alvaro Ganzhez- Jonathan Godwin, Tobias Pfaff, Rex Ying, Jure Leskovee, and Peter W. Battaglia. Learning to Simulate Complex Physics with Graph Networks, 2020. URL https:
//atxii..rrg/ab//200000 09405
Arne Schneuing, Charles Harris, Yuanqi Du, Kieran Didi, Arian Jamasb, Ilia Igashov, Weitao Du, Carla Gomes, Tom L. Blundell, Pietro Lio, Max Welling, Michael Bronstein, and Bruno Correia. Structure-based drug design with equivariant diffusion models. Nature Computational Science, (12):899-909, Dec 2024. ISSN 2662-8457.
doi: 10. 10110315455888004400077.11
SciPy. Wasserstein distance, 2025.
URL https://boos..ccc...ccoooo//oee
refferenceegeneratee//see wtw.sssstttts.ssssst.......... www.seereeeinnnnateenea.... wasserstein-distaan
Dule Shu, Zijie Li, and Amir Barati Farimani. A physics-informed diffusion model for high-fidelity
flow field reconstruction. Journal of Computational Physics, 478:1 2023, ISSN 0021-9991. doi: aaticcle//1//0002.991220000
URL hetttpa://ww.sseee wttt:///wwwwsseeeeerrreeete...e http:///www.rettt..cccc/ccceeeee sttm///ceenee////
Jiaming Song, Chenlin Meng, and Stefano Ermon. Denoising Diffusion Implicit Models, 2022.
URL htt.:https///arxiiee abspgg/bb/221110 02502
14

---

Published as a conference paper at ICLR 2026
Jan Tauberschmidt, Sophie Fellenz, Sebastian J. Vollmer, and Andrew B. Duncan. Physics- constrained fine-tuning of flow-mater mwwching models for generation and inverse problems, 2025. URL
https://arxiv wwt./bs.////ooooooootttt.tt..att 2508 09156
Nils Thuerey, Konstantin Weißenow, Lukas Prantl, and Xiangyu Hu. Deep learning methods for reynolds-averaged navier-stokes simulations of airfoil flows. AIAA Journal, (8(1):25-2)) 2020.
Alexander Tong. Kilian Fatras, Nikolay Malkin, Guillaume Huguet, Yanlei Zhang, Jarrid Rector- Brooks, Guy Wolf, and Yoshua Bengio. Improving and generalizing flow-based generative models with minibatch optimal transport. Trunsactions Machine Learning Research, 2024. ISSN 2835-8856. URL /ntp::ht///// wwt./ppeeeviii.e
Utkarsh Utkarsh, Pengfei Cai, Alan Edelman, Rafael Gomez-Bombarelli, and Christopher Vincent Rackauckas. Physics-constraamed flow matching: Sampling generative models with hard con- straints, 2025.
http//awwww..rrrrb////......0.0. 04171
Mario Lino Valencia, Tobias Pfaff, and Nils Thuerey, Learning distributions of complex fluid sim- ulations with diffusion graph networks. In The Thirteenth International Conference on Learning Representations, 2025.
C. Villani. Optimal Transport: Old and New. Grundlchren der mathematischen Wissenschaften. Springer Berlin Hcidelberg, 2008. ISBN 9783540710493. URL httttt:///boooee googll........o......o..oo......
St/bookssid=NVA. 1AN.2_GGGNNXNNAAAAA.. NAEACAA]
Sinong Wang, Belinda 7. Li, Madian Khabsa, Han Fang, and Hao Ma. Linformer: Self-Attention
with Linear Complexity, 2020. UtLhttps::// http://axxvvv... www.oorgggabbgooooeeee abw//006...... 04768
Haixu Wu, Huakun Luo. Haowen Wang. Jianmin Wang, and Mingsheng Long. Transolver: A fast transformer solver for PDEs on general geometries. arXiv preprint arXiv:2402. 023666 2024.
Zhanhong Ye, Xiang Huang, Leheng Chen, Hongsheng Liu, Zidong Wang, and Bin Dong. PDEformer: Towards Foundation Model for One- Dieeennnnii Partial Differential Equa- tions. In ICLR 2024 Workshop on Al4DDifferenriallquattonn In Science, 2024. URL hetps: //openrevi ew. net. fort/forum? dGGT.DCCW
Mingze Yuan, Pengfei Jin, Na Li, and Quanzheng Li. PIRF: Physicsinnformed reward fine-tuning for diffusion models, 2025.
http://wwwwwxccaa////aaaaaa 20570
Yi Zhang and Difan Zou. Physics-informed distillation of diffusion models for PDE-constrained generation, 2025. URL httl:/tpee/arrvv.oorr//bbbb///// .2391
Hang Zhou, Yuezhou Ma, Haixu Wu, Haowen Wang, and Mingsheng Long. Unisolver: PDE- Conditional Transformers Are Universal Neural PDE Solvers. In ICLR 2025 Workshop Foundation Models in the Wild, 2025a. URL https:/ http:s:///opnreeeww...o. http:/www.ee...teeeeeeeeee/ ffrtt//formmiee
6H1gUgkimm
Zihan Zhou, Xiaoxue Wang, and Tianshu Yu. Generating physical dynamics under priors. In The Thirteenth International Conference on Learning Representations, 2025b. URL https:
//peeneeiew. wwtp://tteteeeeeeeeeeeeee
6COH
Yinhao Zhu and Nicholas Zabaras. Bayesian deep convolutional encoder-decoder networks for surrogate modeling and uncertainty quantification. Journal of Computational Physics, 366:415-
447, 2018 ISSN 0021-9991. doi: http://doo.ao////ooooojjjgpozooa URL https:
www.sccienee.rret..coom www.:ccceeeeee.c.oocacccrrrrrare
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USE OF LARGE LANGUAGE MODELS
Large language models were used exclusively for text editing and correction. They were not involved in the generation of ideas, analyses, substantive content.
A FLOW MATCHING
Flow matching has recently gained attention as a compelling alternative to diffusion models, offer- ing a more direct and computationally efficient framework for generative modeling (Lipman et al.) 2023). It enables high-quality sample generation with significantly fewer function evaluations (Lsser et al. 2024). Given a known source distribution p and an unknown target distribution q, flow match- ing learns vector field uh, parameterized by neural network, that generates probability path pt interpolating from po = p to pt = 4 (Lipman et al. 2024). The learning objective is defined as:
(0) Ex-2( (0,1), x-op. u/ (x) (1))ll
where 9 denotes the model parameters. While multiple formulations exist for the target velocity field We, particularly simple and effective one leverages optimal transport (OT) (Tong al. (2024). In this setting, samples from the base distribution = (0, are linearly transported to via the conditional flow:
(1 min x
ua(2r mn)) 1 (1 amin
with min 10-3, and the corresponding interpolant:
ex( = (1 - (1 min fxr
This yields a straight-line conditional flow with a time-independent vector field. Sampling from the trained model requires integrating the learned field over time:
T1 [ (xx) dt,
typically using numerical ODE solvers such as Euler's method. Although the true OT vector field is constant, the learned approximation typically is not, and integration quality still depends on the time discretization.
B ABLATION WITH TOY PROBLEM
To provide intuition for the proposed improvements, consider ablation with toy problem where the neural network outputs are the and coordinates of points on circumference subject the physical constraint of unit radius. Fig. 7 shows the resulting point distribution along with the corresponding absolute physical error through mean estimation of the final sample. We compare with diffusion models DM, as a DDPM representative (Song et al. 2022), and PIDM-ME, the best performing physics-informed algorithm (Bastek al.) (2025), and variant of the flow matching approach that uses Config without unrolling. It is apparent that each extension yields noticeable gain, and the final PBFM model exhibits residual error that is 61.8 and 27.5 times lower than the DM and PIDM-ME baselines.
C DATASET GENERATION AND RESIDUAL COMPUTATION
We provide detailed description of the datasets used and generated, as well as the method employed to compute residuals during training.
C.I DARCY FLOW
We use the dataset introduced by Bastek (2024) to enable direct comparison with their results. For completeness, briefly summarize the key characteristics of the dataset. The underlying physical model governed by the steady-state Darcy equations, which describe fluid flow through a porous medium:
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DM
PIDM-ME
FM-OT- Logit Storhastic 10
RaB
FM-OT-Uaifom Uniform Deterministic
PM-OT- Utoloon-toociustie
MM-OOT-Logt-DEE Determimistic
10-5
ON--OT-Logii-Soonn
0
E1
Figure 7: Point distribution and absolute error of the physical residual (circle radius squared) for SOTA reference DM, PIDM-ME (Bastek et al. 2025), and all proposed approaches.
u(x:) = Vp(x), x E
u(x)) f(x), x
Ma(m) = 0, E 82
p(x) da =
Here, f(x) denotes the unit outward normal on the boundary 02. The source term f(x) is defined
if x4 1 < w
f(x)
r if 2i 1+ otherwise
f S f
with 10 and w 0.125. The permeability field K (x)) is modeled as K(x) ex))) where G(x)) a Gaussian random field. The pressure field is generated by solving least-squares problem based on 64-term truncated Karhunen-Loèva expansion, following the approach of Jacobsen et al. (2025).
To ensure consistency and avoid introducing numerical discrepancies, we adopt the same procedure for computing residuals during training as was used during dataset generation. Specifically, we em- ploy identical finite difference stencils implemented via 2D convolutional layers, and apply the same reconstruction method for the forcing term f. The pressure field also normalized by removing the integral contribution.
C.2 KOLMOGOROV FLOW
We generate two distinct datasets (training and validation) for the Kolmogorov flow problem with Reynolds numbers in the range [100, 500), using spatial resolution of 128 128. The simula- tion is based on the vorticity-stream function formulation. The velocity field is obtained from the computed vorticity through the stream function, while the pressure field is derived by solving the pressure Poisson equation. We employ TorchFSM (Liu et al. (025b) to perform GPU-accelerated flow simulations using the Fourier spectral method. The training dataset includes
different flow conditions sampled via Halton sequence (Kocis & Whiten 1997), while the validation dataset contains 16 conditions. For each condition, 256 simulations are conducted with slightly perturbed initial states. Simulations are run for 10 000 time steps to reach statistically steady state, followed by data sampling every 4 000 time steps. With a time step size of dt 1/Re, this yields 024 snapshots per condition.
To ensure consistency, the divergence of the velocity fields is computed using the same numerical scheme the spectral solver employed for dataset generation.
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C.3
DYNAMIC STALL
The dynamic stall datasets used for training and validation are generated by solving the unsteady, compressible. vro-dimensional RANS equations around sinusoidally pitching NACA0012 airfoil. An O-grid mesh is utilized, featuring 512 nodes along the airfoil surface and 128 nodes in the normal direction. Simulations are performed using ANSYS Fluent 2024R2 (ANSYS 2024). The governing equations are discretized using second-order upwind scheme for spatial accuracy and second- order implicit scheme for time integration. Gradient reconstruction employs a least-squares cell- based method, while fluxes are computed using the Rhie-Chow momentum interpolation. Pressure- velocity coupling is handled through coupled solver. Airfoil pitching is modeled by prescribing a rigid-body motion to the entire mesh, defined by the angular velocity function dx(t) a, cos(wt). The mean angle of attack, ag. is applied at the start of each simulation by rotating the mesh ac- cordingly. To close the RANS equations, the SST turbulence model with intermittency transport equation is employed. Each oscillation cycle is resolved using 2 048 time steps, and simulations are run until periodic convergence achieved (Baldan & Guardone 2024 2025).
Table Range of conditioning inputs that define the operating conditions of the pitching airfoil. Max
Variable
Min
Mach
0.3
0.5
5
10"
O's
5
10"
k
0.05
0.1
The design space is defined as a four-dimensional hypercube, with each axis corresponding to a conditioning input for the neural network. These inputs include the free-stream Mach number, the mean angle of attack ranges for each variable are provided in Table 7 Training and testing datasets are constructed using Halton sequences. The hypercube is sampled with 128 points for training and 16 points for testing. Each sampled point represents nominal operating condition. Each nominal condition is perturbed as follows:
the pitching amplitude and the reduced frequency k = we/2V The
Epeertureed = (1 + (0, 0.02)) Tnominal
where (0, 0.02) denotes a Gaussian noise term with zero mean and standard deviation 0.02. This results in 32 perturbed variations per nominal condition, yielding total of 128 32 simulations for training and 16 x 32 for testing. To reduce computational costs, all simulation fields are downsam- pled to a resolution of 128 x 128. The saved quantities include fields of absolute pressure, density, temperature, signed skin friction, and tangential velocity gradients across the airfoil surface over full pitching cycle. Fig. illlustrates a spatio-temporal representation of the wall shear stress (Tw) across one complete pitching cycle, highlighting the pitch-up and pitch-down phases and demon- strating the structured mapping of surface data over time.
D ADDITIONAL SAMPLES AND ANALYSIS
We conclude the analysis of the proposed test cases by presenting additional comparisons and com- plete field prediction examples.
D.1 DARCY FLOW
Fig. Iisows the pressure and permeability fields for the FM-OT and PBFM methods under analysis. To ensure fair comparison of the residuals, all fields have similar variable ranges, since higher magnitudes can lead to disproportionately large residual values. This choice is further justified by the residual plots, which exhibit higher absolute emrors in regions with large permeability. Despite local differences, a clear difference emerges: the FM baseline presents residual peak values around 100, while PBFM method peak residuals are reduced by approximately order of magnitude.
To conclude our analysis of the Darcy flow, Fig. I0 compares the DiT architecture with the UNet used by Bastek et al. (2025). This comparison corresponds to the FM setup without residual loss. In
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Pitch-up
Pitch-down
Min
Max
Min a
Trailing
Edge
Suction Side
Leading
Edge
Pressure
Side
Trailing Edge
0.0
t/T
400
200
200
Figure 8: Example of spatio-temporal contour of the post- post-processed Tu distribution over an entire pitching cycle.
Permeability K
Pressure
RMAE K.
0.2
0.1
FM-OT
0.0
100
0.1
10a
0.2
PBFM (Ous))
0.1
10
0.0
10a
-0.1
10-1
Figure 9: Darcy flow examples of pressure and permeability fields along with the physical residual for the proposed approaches.
the first panel, which reports the FM-OT loss during training, we observe that the UNet architecture reproduces the overfitting behavior noted by Bastek et al. (2025). In contrast, the DIT architecture exhibits stable and monotonic reduction in training loss. The second panel shows the residual error, which remains comparable between the two models overall, with the exception of slight dis- crepancies at one function evaluation. Finally, in the third panel, the Wasserstein distance reveals a clear gap in performance: the UNet consistently incurs higher errors, with the distance being ap- proximately twice large for pressure and up to four times larger for permeability, underscoring the superior distributional accuracy of the DiT model. Additional samples produced by our method shown in Fig. 11
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10
10
RasE
10-
[
10
10
DiT Log-L Normal-Stee Stoduastic
0.0
0.5
1.0
10r
10"
Trainin iterations
x10
stepe
UNet-Logt Normal Stodhastic
Wi(P)
W. ((KK)
10
FM FM steps
FM steps
Figure 10: Comparison of UNet (Battek et al, 2025) and DiT architectures Darey flow, The physical residual (error bars refer to min-max values within the validation dataset samples) and Wasserstein distance, as a function of FM steps, are computed over 1024 samples.
Pemmakiity
Preseum
0.5
0.2
0.0
0.00
-0.25
Figure 11: Representative Darcy flow field samples generated with PBFM using 20 FM steps. The samples illustrate the diversity in flow behavior, particularly highlighting variability in the range extrema.
D.1.1 NON-DIMENSIONAL SCALING
Even though the role scaling well recognized in machine learning, its impact on physical residuals and distributional performance remains important aspect examine. We study this effect in the context the Darcy flow problem by comparing two scaling strategies. In the first approach, each variable is normalized to have
mean, yielding the following scaling values: Pmean = 0.0, Perd 0.576, kmean = 1.386, kad 10.64. The corresponding results are reported in Table The second approach, denoted the Scale version, rescales the pressure p the range [- and the permeability & to [0, 1]. The results for this setting are shown in Table g Across all methods, this alternative scaling consistently degrades performance, leading higher physical residual errors and poorer distributional metrics.
D.2 KOLMOGOROV FLOW
Fig. 12 illustrates example predictions from the FM-OT, and PBEM models, along with their cor- responding physical residuals, representing the divergence of the predicted fields. The baseline FM
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Table 8: Generative performance metrics for Darcy flow problem 1024 samples. RE: physi- cal Residual MSE, WD: Wassserstein Distance, JS: Jensen-Shannon divergence, NFE: Number of function evaluations, IT: Inference wall-clock time. Best and second best results.
Dataset Metric
PBFM Scale FM-OT Scale DiffusionPDE Scale
D-Flow Scale?
RE WD -102 JS -10
2.260
5,551
4.307
2,417
0.185
0.076
0.075
0,437
Darcy
0.218
0.152
0.142
0.353
NFE IT [s]
20
20
20
20
0.101
0.100
0.590
3.126
$ D-Flow method unstable, samples are filtered using condition resulting in 403 valid ones.
exhibits residual MSE the order of 10-1 across most of the domain, whereas the unrolled framework significantly reduces the error, dropping below 10-3 large regions of the field, despite some localized areas with higher residuals.
a-velocity
velooccity
RASSE
10-1
PW-OT
10
10
10
PBFM (Onis)
10
10
Figure 12: Kolmogorov flow example of u- and v-velocity fields and physical residual, divergence, MSE for the proposed approaches using 20 FM time steps.
Furthermore, Fig. 13 shows the mean and standard deviation of the predicted fields for FM-OT and PBFM models, alongside the reference data. The frameworks closely match the reference mean, with the unrolled version providing a smoother, less oscillatory solution compared to FM-OT. Re- garding the standard deviation, none the models fully capture the reference distribution, partic- ularly missing the peak value of approximately 0.95, although their performance remains broadly comparable.
Additional samples produced by PBFM method are shown in Fig. 14
D.3 DYNAMIC STALL
For the dynamic stall case, Fig. 15 presents the mean and standard deviation of the predicted fields obtained using PBFM. The predictions show strong agreement with the reference data, both in terms of overall distribution well capturing the extrema of each variable. The only notable deviation appears in the standard deviation of the skin friction, which slightly exceeds the corresponding reference values.
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Reference
FM-OT
PBPM (Ous)
Mean
0.5
0.0
0.5
0.5
0.0
0.5
-0.5
0.0
0.5
Std
0.8
0.9
0.6
0.7
0.8
0.6
0.7
0.8
Figure 13: Example of mean and standard deviation of Kolmogorov flow computed over 20 FM steps with 128 samples for the proposed approaches.
u- velocity
2
2
2
0
e- velocity
Figure 14: Example of Kolmogorov flow field samples generated with PBFM using 20 FM steps.
Fig. 16 comparess the performance of the proposed models as a function of the number of FM integra- tion steps. The first panel focuses on the physical residual MSE, indicating that the best performance achieved with 10 FM steps, with PBFM consistently delivering one order of magnitude reduction compared the baseline. The second panel reports the Wasserstein distance, showing that PBFM also narrows the gap with the reference data distribution, reaching its optimal value at 20 FM steps. Finally, the MSE of the mean and standard deviation reveals that the proposed methods preserve the already strong performance of the baseline, introducing only minor deviations.
Additional qualitative samples produced by our trained PBFM method are shown in Fig. 17 They highlight the wide range of physical behavior modeled by this case: from small oscillation attached flow at the top, to deep dynamic stall cases with shock wave formation at the bottom.
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de
jay
Roferenes Meen
50000
1000000
D
0
2
250
300
0.5
1.0
250
0
250
x100
x107
PEEM Meon
50000
100000
250
300
0.5
1.0
250
0
250
x10
X10
Redereecc Std.
25000
2.5
5.0
0.0250.00000755
x10
x100
PRFM Sul
2500 5000
2.5
3.0
1
3
3
20
0.125005500.00
50
100
x10
Figure 15: Example of mean and standard deviation of dynamic stall problem computed over 128 samples with 20 FM steps.
10-1
10-5
10-3
Ryst
10-6
10
10
10P
10
10P
10a
10
10
McarAsB
Stdyeen
10-5
10-5
10a
10a
FM steps
FM steps
FM-OT
FM-OT xx with Config
PRFM
Figure 16: Comparison of the proposed approaches for physical residual and Wasserstein distance, mean and standard deviation MSE as function of FM steps for dynamic stall case. Error bars refer to the different conditioning values within the validation dataset.
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P
du do
du/dy
50000 100000
-250
250
"100
x100
Figure 17: Example of dynamic stall flow field samples generated with PBFM using 20 FM steps. The outputs vary substantially depending on the conditioning inputs, illustrating the model's sensi- tivity to different flow scenarios.
RESIDUAL LOSS SCALING LAWS
During training, the residual of each sample is computed starting from given time and evolved forward until t = 1. Trajectories initialized at times closer to 0 tend to exhibit larger errors, particularly when only a single integration step is used. To mitigate this effect, we introduce weighting scheme based on power law t?, where the residual loss is scaled according to the starting time t. We investigate the impact different power exponents p, focusing on the most challenging case of dynamic stall.
Fig. 18 presents a comparison of the MSE for physical residuals, as well as the mean and standard deviation, for both the FM-OT with Config and PBFM frameworks. The results show that unrolling helps regularize the error, producing monotonic increase in error as a function of the power p. and also reduces sensitivity the choice of p in the range [1, 4]. Notably, both frameworks achieve optimal performance when residuals are scaled linearly with time, p = 1. In contrast, using no scaling, p 0. results in significantly higher errors, underscoring the importance of appropriately weighting residuals based on the starting time.
F CONFLICT-FREE UPDATES AND WEIGHTED LOSS TERMS
The introduction the second loss term associated with the physical residual minimization trans- forms the framework into multi-task learning problem. A seemingly attractive approach is to combine the two losses using a fixed weighting byperparameter. However, this naive strategy re- quires manual tuning of the relative weight and often leads to suboptimal performance. In particular, optimization may get stuck in a local minimum of one loss due to conflicts between the tasks. To ad- dress this, adopt the conffict-free updates (Liu al. 2025a). which mitigate gradient conflicts by
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FM-OT LR with Config
14
1.4
X10
6.5 X10
1.2
Meanusa
1.2
Rons
Staue
5.5-
1.0
0.8
5.0-
0.8
lyperparaneter
Lx
P--R
14 x10
6.5
x 10
6.0-
PDFM
RAss
Mesanase
SkdusE
5.5
1.0
5.04
hyperperameter Cx
Figure 18: Comparison of physical residual, mean and standard deviation MSE for dynamic stall under various power-law scalings in the residual loss. Unrolling reduces sensitivity to the scaling exponent.
computing a non-conflicting optimization direction through the inverse of the loss-specific gradient covariance matrix. Furthermore, this approach has the potential to yield improved solutions.
We evaluate the proposed setup on the dynamic stall case, the most challenging scenario considered, comparing ConFIG models trained with various fixed loss weights. Fig. 19 reports the MSE for the physical residual, the predicted mean, and the standard deviation across all configurations. ConFIG consistently outperforms the fixed-weight approaches, achieving the lowest error in each individual metric and delivering the best overall performance.
To further assess the effectiveness of ConFIG, quantified gradient conflicts via pairwise cosine similarity between physical and FM losses. For the best model using 4-step unrolling with ConFIG, the average conflict is 6.93%, whereas model with fixed residual scaling at 500 exhibits much higher average conflict of 19.98%.
G ARCHITECTURE AND TRAINING DETAILS
The framework is implemented in PyTorch v2.5.1, 1, employing Distributed Data Parallel (DDP) for scalable training. All experiments are trained using the AdamW optimizer with weight decay set to 0, 31 0.5, 82 0.999, and a fixed learning rate. To avoid learning rate adjustments across different hardware setups, we maintain a constant global batch size, independent of the number of GPUs. An Exponential Moving Average (EMA) of the model parameters is maintained throughout training, with decay rate of 0.999, and is used during sampling.
We adopt the Diffusion Transformer (DiT) architecture proposed by Peebles & Xie 2023 as the backbone for our flow matching model, incorporating minor modifications. The model is condi- tioned via adaptive layer normalization (adaLN-Zero) blocks, which replace the standard normal- ization layers. The scale and shift parameters in these blocks are derived from the sum of the em- bedding vectors for the time step I, used in the flow matching process, and the conditioning signal
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10
Meanuee
Raese
10
10-91
@ hyperparanzeer + Lyxt +
Cn
50
100
1000
2000
ConFIG
Figure 19: Comparison of MSE for physical residual, predicted mean, and standard deviation in the dynamic stall case, using ConFIG and fixed hyperparameters for loss weighting. The optimal configuration minimizes the across all three metries simultaneously.
C. We introduce two modifications compared to the original DiT implementation. First, we incor- porate linear attention (Wang et al. 2020) alongside the standard quadratic one. Second, replace the original label embedder with two-layer module: a linear layer followed by a SiLU activation function, and second linear layer that produces the final conditioning emhedding.
All the hyperparameters are summarized in Table
Table 9: Architecture and training hyperparameters for proposed test cases.
Darcy flow Kolmogorov flow
Dynamic stall
Training iterations
1248000
512000
2048000
Learning rate
3 3-10-5
1- 10
1 10
Batch size
64
64
64
Conditioning size
1
Output size
2
Patch size
4
Hidden size
256
256
128
DiT depth
8
Attention heads
8
8
Attention type
Quadratic
Linear
Linear
Parameters (M)
9.75
9.81
1.29
Gflops
2.72
13.00
1.68
H PERFORMANCE EVALUATION
Table TO reports the inference wall-clock times for generating 32, 64, and 128 dynamic stall samples using CPU cores an Intel Xeon Platinum and an NVIDIA A100 64GB GPU. The same CPU was used to run the numerical simulations, ensuring more consistent comparison of computational performance. On average, a single numerical simulation requires 76 minutes to complete. In con- trast, the proposed generative model can produce 128 samples in just 2 and minutes when using 10 and 20 FM steps on the CPU, respectively. When executed on a modern GPU, these inference times drop dramatically to 0.2 and 0.4 seconds, respectively. This substantial speedup highlights the model's s potential as fast and reliable surrogate for dynamic stall prediction in helicopter and wind turbine applications.
I SAMPLER IMPLEMENTATION AND ADDITIONAL DETAILS
Algorithm outlines the proposed sampling procedure, which is implemented using the explicit Euler integration scheme with n equispaced time steps. The process begins by initializing z,, which holds the sample values time 0, with Gaussian noise. During the integration loop, each time
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Table 10: Wall-clock time in seconds to generate samples using 8 cores of an Intel Xeon Platinum and an NVIDIA A100 64GB for dynamic stall case. Batch size is set equal to the number of samples. One simulation with the same cores takes on average 76 minutes (4.56 10 FM steps 32
Number of samples GPU
Number of samples CPU
64
128
32
64
128
1
3.23
10
3.38
10-3
3.80
10-3
2.27
10
6.16
10
1.20
10a
2
9.76
10-3
1.50-10-2
2.53
10-2
4.48
100
1.11 11
2.40
10a
5
2.90
100
3.99-10"
1.17 10
10¹
5.99
10a
10
6.15-10-2
1.07-10-1
1.98-10-1
2.25 10
- 10t
1.19-10
20
1.26
2.23-10-1
4.13-10-1
4.68
10a
1.15
2.40 10a
50
3.21 10-1
10-
1.06 10
1.12 10
3.08
5.99
10
step can be computed using either the standard deterministic FM sampler or the proposed stochastic variant. The choice between the two is governed by user-defined boolean control parameter and is restricted to the initial segment of the trajectory, up to a threshold time t*. In our experiments, we set t" === 0.2, introducing additional stochasticity during the early phase of sampling while preserving high sample quality in later stages. the stochastic sampler, the velocity u, is used to generate the final sample in single forward step, followed by backward update to time t dt using Gaussian noise sample.
Algorithm Deterministic and Stochastic Samplers dt < 1/n Te + 3o 10,
D is the number of integration steps
for i = 0, < n do
if and use stochastic sampler then
f- + (1 .uf
D. Integrate to /
{ + dl
< (1 - NN + 3
Return to + dt using new generated normal noise Standard deterministic sampler, stochasticity is embedded in zo
else
3, Zz + dt ul
t t + dt
end if end for
Our method introduces some bias during training, affecting the pure deterministic sampler (t* 0.0) and resulting in higher distributional error. However, for the stochastic sampler, the observed decrease in residual error and increase in distributional accuracy is also present for vanilla FM, indicating that this effect not unique to our approach. The difference in distributional error in the deterministic sampler primarily arises due to conflicting gradients during training: when physical constraints and distributional objectives are jointly optimized, their gradients can oppose each other, leading to suboptimal updates for one or both objectives.
Table Performance of stochastic sampler for Darcy flow problem over 1024 samples using 20 FM steps using FM-OT. Noise resampling is enabled for t < t², details in Algorithm 2 t* 0.0 coincides with deterministic sampler.
t"
Metric
0.0
0.2
0.4
0.6
0.8
1.0
RE
4.159
3.530
3.010
2,581
2.018
1.174
WD -102
0.059
0.102
0.123
0.255
0.403
0.403
JS 10"
0.131
0.197
0.246
0.324
0.414
0.436
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Table 12: Performance of stochastic sampler for Darey flow problem 1024 samples using 20 FM steps using FM-OT+ConFIG with residual loss. Noise resampling is enabled for t < t², details Algorithm
0.0 coincides with deterministic sampler.
Metric
0.0
0.2
0.4
0.6
0.8
1.0
RE
1.331
1.448
1.317
1.152
1.002
0.765
WD -022
0.391
0.096
0.139
0.254
0.383
0.361
JS 10
0.293
0.193
0.266
0.320
0.395
0.404
To provide complete picture, report results for vanilla FM (Table [1] and for FM with the addition of physical loss but without unrolling (Table 12). complementing the unrolling results presented in Table 3 The stochastic sampler helps prevent collapse "unique" solutions, a phe- nomenon typical of PINN setups where deterministic optimization can lead to mode collapse and reduced sample diversity. By leveraging stochastic sampling, our method maintains higher distribu- tional fidelity.
J PBFM VARIANTS
We performed two ablations on the dynamic stall benchmark to assess whether decomposing the physics objective or modifying gradient flow could yield further gains. The evaluated variants are:
PBFM 3 losses: the physics objective is split into two separate residual losses, Rig and R.,. trained alongside the FM loss.
PBFM last step: of the unrolled predictions when computing the residual loss to reduce backward memory and computational time.
the base PBFM but the gradient is computed only at the last step
Table 13:
Comparison of wall-clock time in seconds for training iteration and memory usage in GB on an NVIDIA A100 64GB GPU for the proposed approaches. Batch size 64 for all cases. Inference time is unchanged. Method
step
2 steps
steps
steps
PBFM
[s]
8.14
10-2
1.18
10
1.55
10-1
1.90
10-1
[GB]
4.41
7.18
9.90
12.58
PBFM
[s]
1.08
10
1.68
10
2.29 10-1
2.84
10
losses
[GB]
4.41
7.20
9.91
12.59
PBFM
8.35
10-2
1.09
10-1
1.23
10-1
10-1
last step
[GB]
4.41
8.48
8.48
8.48
Table 14: Generative performance metrics for dynamic stall problem using 20 FM steps. RE: phys- ical Residual MSE, WD: Wassserstein Distance, JS: -Shann-h divergence, MMSE: MSE of the mean fields, SMSE: MSE of the standard deviation fields. Dataset Metric
PBFM PBFM 3 losses
PBFM last step
RE -106
0.339
0.315
0.392
WD 1
1.814
2.077
3.038
Dynamic IS -
0.680
0.653
0.722
Stall
MMSE -10
1,490
1.696
1.814
SMSE 1
0.874
0.842
0.809
Splitting the physics terms into separate losses does not bring clear overall improvements. The three-loss variant attains slightly lower residual
Key findings are summarized in Tables 13 and 14
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MSE in our runs, 0.315 vs. 0.339, while the Wasserstein distance increases from 1.814 to 2.077. Moreover, the three-loss setup increases per-iteration training time by roughly 30-40%, due to ad- ditional gradient computations: peak memory usage increases only marginally. Aggregating the physics constraints into single residual loss and resolving conflicts via ConFIG provides the best trade-off between aceuracy and training cost for the dynamic stall case.
The last step variant reduces the backward computation costs while the impacts on final accuracy limited. We also measured gradient alignment between the two physies terms and found them largely aligned (average cosine similarity s 0.76), which explains why aggregating the physics terms into a single loss is both effective and more efficient practice.
29


Reviews:
[{"review_id": "a4RGXOdXl2", "reviewer": "Reviewer_e5nq", "summary": "This paper proposes a framework that use flow-matching to generate admissible solution for PDEs. The key feature of the paper is that it recongize the potential gradient conflicts between the distribution fidelity and the physical consistency and propose a way to obtain the Pareto optimal solutions. Across three benchmarks—Darcy (steady), Kolmogorov (divergence-free), and a challenging dynamic stall aerodynamics case with analytic constraints—PBFM improves the physics–vs–distribution trade-off and maintains FM-like inference cost (e.g., competitive wall-clock and NFE), while acknowledging higher training memory/time due to unrolling and an extra backward pass. Figures and tables report consistent residual drops, better Wasserstein/JS metrics than inference-time constraint baselines, and near-FM inference speed; a discussion contrasts training overheads with the gain in accuracy and shows large end-to-end speedups over classical solvers once trained. The study is quite completed and the problems the authors selected for numerical experiments are chosen carefully.", "questions": "Can you split the boundary and residual loss and report the gradient conflicts among these physical losses and the distribution fidelity loss?\n\nCan you gate which physics term to optimize per step (or reweight it) using online conflict indicators (e.g., negative cosine, norm ratios) or a simple KKT/dual ascent scheme, and show coverage–fidelity trade-offs?\n\nWhat extra memory/backprop cost does BC/divergence decomposition incur vs. the single physics loss?", "limitations": "", "rating": 10, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "It makes sense to cast “physics vs. distribution” as a true multi-objective problem and use conflict-free gradient updates (ConFIG), so that each step descends both the flow-matching loss and the physics residual, thereby avoiding brittle manual weights and consistently improving the Pareto front over fixed-weight baselines.\n\nIt mitigates Jensen's gap through training-time unrolling, thereby lowering the residual errors without incurring additional inference costs. \n\nI appreciate that the authors use numerical examples closer to engineering applications (Darcy, Kolmogorov, and dynamic stall) beyond the standard benchmarks (Poisson's equations and Burger's equation), etc.", "weaknesses": "Physical residuals are sensitive to the minimum noise set as a hyperparameter;​ higher noise degrades residuals and requires careful tuning. The method also introduces a time-based residual scaling, whose choice affects errors (albeit less so with unrolling). Some elaborations on how to handle the hyperparameter sensitivity would be helpful. \n\nThe paper aggregates boundary-condition penalties and the divergence-free residual into a single “physics” loss. However, these terms can induce conflicting gradients (e.g., negative cosine similarity), leading to oscillatory updates or favoring one constraint over the other.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Z39tA9NMuB", "reviewer": "Reviewer_EdS2", "summary": "The paper studies physics-constrained generative modeling and introduces a method that enforces physical constraints during training using conflict-free gradient updates and unrolling. This approach avoids manual loss balancing, improves generative fidelity, and enables efficient inference. The paper also presents numerical evidence supporting the effectiveness of the proposed method.", "questions": "I think the authors could include a figure illustrating the concept of conflict-free gradient updates. This would help a general audience better grasp the idea and make the content more accessible.", "limitations": "", "rating": 4, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- The paper addresses a fundamental problem in generative modeling: how to enforce known physical constraints to improve the fidelity of generated data.\n\n- The proposed method requires more memory storage and training time, which might not scale well for larger systems.\n\n- The paper clearly identifies the challenge of conflicting gradient updates and introduces a corresponding method that enables gradient descent to simultaneously reduce both the matching loss and the physical constraint loss.", "weaknesses": "-The proposed method appears somewhat straightforward, as it primarily leverages an existing approach (ConFIG) to compute conflict-free gradient updates.\n\n- Section 3.3 is a bit difficult to follow. The authors could add a figure to illustrate why this issue needs to be addressed and to clarify the concept of Jensen’s gap. In particular, is this a typo - should the final clean sample be $x_1$ rather than $x_0$?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "aNJuxqLzFL", "reviewer": "Reviewer_XbPf", "summary": "The paper introduces \"Physics Based Flow Matching\" (PBFM), a method that treats the tension between physical consistency and generative fidelity as a Pareto optimization problem. By combining conflict-free gradient updates, residual unrolling, and stochastic sampling, PBFM claims to achieve a Pareto-optimal trade-off between residual accuracy and sample quality across PDE benchmarks, without a significant increase in inference cost.", "questions": "See Weaknesses above.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "S1. The empirical study is comprehensive, including ablations on unrolling, stochastic sampling, and Gaussian noise effects, which clarifies the contribution of each component and isolates the respective observed gains.\n\nS2: The experimental benchmarks are well chosen and physically meaningful, spanning PDEs with increasing complexity. The results demonstrate stable improvements across tasks, and the method manages to maintain minimal inference overhead despite additional training components.\n\nS3: Writing is clear and easy to follow.", "weaknesses": "**W1.** Many components of the proposed framework: residual-based losses from PINNs (Raissi *et al.*, 2019), **ConFIG** gradient orthogonalization [4], and stochastic sampling from ECI [1] are adapted rather than novel. The main contribution lies in integrating these existing mechanisms under a Pareto framework rather than introducing fundamentally new algorithmic principles. Moreover, since PBFM enforces *soft* constraints, it cannot match the strict physical consistency of hard-constraint approaches such as ProbConserv, ECI, PCFM, and conservation-respecting models [6], which exhibit stronger residual convergence and physical invariance guarantees.\n\n**W2.** Methods like PCFM [2] and ECI [1] already demonstrate superior trade-offs between physical residual and generative accuracy, achieving lower MMSE and FPD (aligned with the Wasserstein metric used here). These approaches explicitly enforce hard constraints throughout training and inference, leading to simultaneous gains in both physical and distributional metrics. In contrast, PBFM’s ConFIG balancing produces only a single equilibrium solution along this trade-off, without exploring or controlling the broader Pareto front. While ConFIG offers one way to mitigate gradient conflict between physics and generative objectives, I assume PBFM adopts it primarily to avoid the conflicting gradient behavior observed in PINNs. However, without broader comparisons, theoretical justification, or reference to recent developments in conflict-free training [7], it remains uncertain whether this choice is optimal within the Flow Matching framework. This uncertainty is compounded by the lack of sufficient empirical evaluation across Pareto trade-offs, leaving the “Pareto-optimal” characterization only partially substantiated.\n\n**W3.** There should be some quantitative comparison to prior physics-aware diffusion and flow methods [1, 2, 3, 5]. The explanation that ECI [1], PCFM [2], and PIDDM lack public implementations or cannot handle overlapping residuals is not convincing. Public code availability is not a sufficient reason to omit benchmarking; if previous works report standardized results, at least partial or analytical comparisons should be provided. For example, **ECI** has an open implementation and published results on the *Heat* and *Navier–Stokes* PDEs. Even if PBFM targets showcases results on different PDEs, evaluating it on these established datasets would contextualize its improvements. As written, the analysis offers only a partial view of progress relative to existing literature.\n\n**W4.** The claim of being “gradient-free” is not strictly accurate, as the method still differentiates through residuals during training. Additionally, the unrolling step introduces extra back-propagation and memory overhead, thereby reducing one of flow matching’s main advantages: its computational simplicity, and without requiring backpropagation through an ODE solve. \n\n\n**References**\n\n[1] Cheng, C., *et al.* “Gradient-Free Generation for Hard-Constrained Systems (Extrapolation–Correction–Inference).” *arXiv preprint* arXiv:2412.01786 (2024).\n\n[2] Utkarsh, U., *et al.* “Physics-Constrained Flow Matching: Sampling Generative Models with Hard Constraints.” *arXiv preprint* arXiv:2506.04171 (2025).\n\n[3] Christopher, J. K., S. Baek, and N. Fioretto. “Constrained Synthesis with Projected Diffusion Models.” *NeurIPS* 37 (2024): 89307–89333.\n\n[4] Liu, Q., Chu, M., and Thuerey, N. “ConFIG: Towards Conflict-Free Training of Physics-Informed Neural Networks.” *arXiv preprint* arXiv:2408.11104 (2024).\n\n[5] Yao, J., Mammadov, A., Berner, J., Kerrigan, G., Ye, J. C., Azizzadenesheli, K., and Anandkumar, A. “Guided Diffusion Sampling on Function Spaces with Applications to PDEs.” *arXiv preprint* arXiv:2505.17004 (2025).\n\n[6] Hansen, D., *et al.* “Learning Physical Models That Can Respect Conservation Laws.” *ICML*, PMLR, 2023.\n\n[7] Wang, Sifan, et al. \"Gradient alignment in physics-informed neural networks: A second-order optimization perspective.\" arXiv preprint arXiv:2502.00604 (2025).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "NXCprFKBXU", "reviewer": "Reviewer_f5gi", "summary": "This paper introduces Physics-Based Flow Matching (PBFM) that balances physical consistency with distributional accuracy. The authors identify the inherent trade-off between optimizing for physical fidelity (via PDE residuals) and generative quality as a core challenge. PBFM addresses this through conflict-free gradient updates using the ConFIG method, eliminating the need for manual loss balancing. Additionally, the method employs unrolling during training to mitigate Jensen's gap—the discrepancy arising when physical constraints are imposed on intermediate predictions rather than final samples. Experiments across three benchmarks demonstrate that PBFM achieves Pareto-optimal performance, outperforming baselines in both physical residual error and distributional metrics while maintaining competitive inference speed.", "questions": "**Question on Stop Gradient for Unrolling.** In Algorithm 1, the unrolled prediction is computed via n integration steps and then used to evaluate the physical residual. This requires backpropagating through the entire unrolling process, leading to the reported memory and training time increase. Have you considered applying a stop gradient operation to before computing the residual? This technique is commonly used in single-step diffusion models and consistency models to reduce computational cost. \n\n**Question on Training-Inference Step Mismatch for Complex PDEs.** The paper demonstrates that the method benefits from unrolling during training and that increasing inference steps consistently reduces residual errors. However, for highly complex PDEs with non-smooth ODE flows, accurate sampling may require substantially more discretization steps (e.g., 50-100 steps). In such cases, training with only a few unrolling steps might fail to generate physically meaningful samples for computing the residual loss, as the partially-evolved states after 1-4 steps could be far from any valid solution. Could the authors discuss: (1) whether this training-inference step mismatch has been observed in practice, (2) how the method would perform when complex PDEs necessitate many inference steps, and (3) whether a curriculum strategy progressively increasing unrolling steps during training could address this issue without making the memory overhead prohibitive? Understanding this limitation would help assess the method's applicability to more challenging physical systems.", "limitations": "", "rating": 4, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "**Novel conflict-free optimization:** The paper identifies the conflicting gradients between generative and physical objectives and proposes an elegant solution using conflict-free gradient updates. This approach is innovative and eliminates the challenging manual tuning of loss weights, enabling simultaneous optimization of both objectives.\n    \n**No inference overhead:** Unlike existing physics-constrained methods that require costly iterative sampling procedures at inference time, PBFM maintains competitive inference speed comparable to standard flow matching, making it practical for real-world applications.\n    \n**Strong experimental results:** The method demonstrates superior performance across three diverse benchmarks, consistently achieving lower physical residuals and better distributional metrics compared to baselines. The Pareto-optimal trade-off and comprehensive ablation studies validate the effectiveness of the proposed approach.", "weaknesses": "**Scalability Concerns.** While the paper demonstrates improvements on 2D problems with modest resolutions (64$\\times$64 and 128$\\times$128), the training overhead raises serious concerns for practical applications. The authors acknowledge that unrolling with 4 steps incurs up to 3$\\times$ memory consumption and 2.5$\\times$ training time. For higher-dimensional problems (e.g., 3D PDEs at 128$^3$ resolution), the computational cost would scale dramatically: (1) storing intermediate states during unrolling becomes prohibitively expensive, (2) computing physical residuals (gradients, divergence) in 3D is substantially more costly, and (3) the ConFIG gradient projection requires additional backward passes. Moreover, more challenging PDEs typically require a larger number of diffusion steps for accurate sample generation, which directly translates to proportionally higher memory consumption during unrolling, potentially rendering the method infeasible. The method's applicability to industrially-relevant high-resolution 3D problems remains unclear, potentially limiting its impact to low-dimensional scenarios where the training overhead is manageable.\n\n**Insufficient Analysis of ODE vs. Stochastic Sampling Discrepancy.** A fundamental principle of continuous normalizing flows is that ODE and SDE samplers should yield samples from the same distribution when properly trained. However, the empirical results in Table 3 show significant distributional differences between deterministic (t* = 0.0) and stochastic samplers (t* > 0), with Wasserstein distance varying from 1.470 to 0.138. This discrepancy suggests that the proposed training procedure may introduce optimization biases that favor one sampling paradigm over the other. The paper claims stochastic sampling as a core contribution (Contribution IV), yet provides no theoretical or empirical investigation into why the training methodology creates this distribution mismatch. Possible explanations include: (1) the conflict-free gradient updates may inadvertently bias the learned velocity field toward deterministic trajectories, (2) the unrolling procedure evaluates residuals on ODE-integrated paths, potentially misaligning with stochastic sampling dynamics, or (3) the residual weighting scheme may not be sampling-invariant. The authors should provide: (1) ablation studies isolating which training components cause the ODE-SDE discrepancy, (2) analysis of whether standard flow matching (without physical constraints) exhibits similar behavior under their training scheme, and (3) theoretical or empirical justification for why their method fundamentally requires stochastic sampling rather than fixing the training procedure to achieve sampling-invariance.\n\n**Extensive Hyperparameter Tuning Still Required.** While the paper claims to avoid manual loss balancing, the method still requires careful tuning of multiple hyperparameters including $\\sigma_{\\text{min}}$, $t^*$ for stochastic sampling, number of unrolling steps, and the power $p$ for residual weighting. Table 5 demonstrates significant sensitivity to $\\sigma_{\\text{min}}$ choice, with performance varying substantially across different values. This undermines the claim of automatic balancing and may limit practical applicability.\n\n**Minors.** Inconsistent use of $x_1$ and $x_0$: In Algorithm 1 and Figure 1, authors use $x_1$ to denote the data, while on line 069, 070, 196, they use $x_0$. Missing reference: Generating Physical Dynamics under Priors; PIRF: Physics-Informed Reward Fine-Tuning for Diffusion Models; Physics-Constrained Fine-Tuning of Flow-Matching Models for Generation and Inverse Problems.\n    \nIf the authors can address the computational scalability to higher-dimensional problems and provide theoretical/empirical justification for the ODE-SDE sampling discrepancy, I would be willing to increase my score.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)