Physics-aligned field reconstruction with diffusion bridge
Authors: Zeyu Li
Hongkun Dou
Shen Fang
Wang Han
Yue Deng
Lijun Yang
Venue: ICLR 2025 Spotlight (2025)
Keywords: Fluid dynamics
diffusion models
super-resolution
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)
TL;DR: We propose a diffusion-based framework for the reconstruction of physical field with enhanced compliance with physical laws.

Abstract: The reconstruction of physical fields from sparse measurements is pivotal in both scientific research and engineering applications. Traditional methods are increasingly supplemented by deep learning models due to their efficacy in extracting features from data. However, except for the low accuracy on complex physical systems, these models often fail to comply with essential physical constraints, such as governing equations and boundary conditions. To overcome this limitation, we introduce a novel data-driven field reconstruction framework, termed the Physics-aligned Schr\"{o}dinger Bridge (PalSB). This framework leverages a diffusion bridge mechanism that is specifically tailored to align with physical constraints. The PalSB approach incorporates a dual-stage training process designed to address both local reconstruction mapping and global physical principles. Additionally, a boundary-aware sampling technique is implemented to ensure adherence to physical boundary conditions. We demonstrate the effectiveness of PalSB through its application to three complex nonlinear systems: cylinder flow from Particle Image Velocimetry experiments, two-dimensional turbulence, and a reaction-diffusion system. The results reveal that PalSB not only achieves higher accuracy but also exhibits enhanced compliance with physical constraints compared to existing methods. This highlights PalSB's capability to generate high-quality representations of intricate physical interactions, showcasing its potential for advancing field reconstruction techniques. The source code can be found at https://github.com/lzy12301/PalSB.

Published as a conference paper at ICLR 2025
PHYSICS-ALIGNES FIELD RECONSTRUCTION WITH DIF- FUSION BRIDGE
Zeyu Li, Hongkun Dou, Shen Fang, Wang Han, Yue Deng, Lijun Yang* Beihang University
(1izeyu123478, dounk, eurekal0shen, drwanghan, ydeng, yangli jun hww.eeeuauuuuuuuaaaaaa edu.cn
ABSTRACT
The reconstruction of physical fields from sparse measurements is pivotal in both scientific research and engincering applications. Traditional methods are increas- ingly supplemented by deep learning models due to their efficacy in extracting features from data. However, except for the low accuracy on complex physical systems, these models often fail to comply with essential physical constraints, such as governing equations and boundary conditions. To overcome this limitation, we introduce novel data-driven field reconstruction framework, termed the Physics- aligned Schrödinger Bridge (PalSB). This framework leverages a diffusion bridge mechanism that specifically tailored align with physical constraints. The PalSB approach incorporates dual-stage training process designed to address both local reconstruction mapping and global physical principles. Additionally, boundary- aware sampling technique is implemented to ensure adherence to physical boundary conditions. We demonstrate the effectiveness of PalSB through its application three complex nonlinear systems: cylinder flow from Particle Image Velocimetry experiments, two- rroddimensional turbulence, and reaction-diffusiio system. The results reveal that PaISB not only achieves higher accuracy but also exhibits en- hanced compliance with physical constraints compared to existing methods. This highlights PaISB's capability generate high-quality representations of intricate physical interactions, showcasing its potential for advancing field reconstruction techniques. The source code can be found at https://gtthub.com/2/ 12301/PaISB.
INTRODUCTION
Field reconstruction critically important in several domains, including fluid mechanics (Fukami al., [2033 Manohar et al.) 2018), metcorology (Carrassi al., 2018 Kondrashov Ghill [2066]
Tello Alonso et al. 2010)) and astrophysics (The Event Horizon Telescope Collaboration 2019).
where high-fidelity data is essential. This process aims recover the spatiotemporal information of
physical system from limited observations gathered by sensors. The challenges of measurement sparsity and noise necessitate efficient and accurate methodologies that enhance understanding of complex systems beyond the resolution capabilities of the instruments used.
However, the intrinsic challenges posed by the ill-posed nature of inverse problems and the complex spatiotemporal interactions within many systems, such as turbulence, render these reconstructions particularly difficult (Buzzicotti) (023). Traditional physics-bare based methods, which repeatedly recali- brate to balance observational data with physical laws, often incur significant computational costs due to their intensive requirements for high-fidelity simulations.
Recent advances in machine learning, especially diffusion-based models, have shown promise in managing complex data distributions and have been successfully applied in diverse areas such as image translation (Zhang al.
2023 Luo al. 2023 Meng al. 2022 Su al. 2023 Yue
et al. 2023), molecular generation (Xu et al. 2022
Watson et 2023 Igashov et al. 2024). and
dynamic forecasting (Gao et al.
2023 Yoon et al. 2023 Cachay et al. 2023). Despite achieving
remarkable point-wise accuracy compared to other end-to-end methods, these models often fail to
"Corresponding author

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Published as a conference paper at ICLR 2025
align with physical laws in the context of physical field reconstruction. The challenge is exacerbated by the non-convex and often intractable nature of the domains defined by the governing physical laws, such as partial differential equations (PDEs). The projection-basee based methods for enforcing hard
constraints (Liu et al., 2023a Lou & Ermon 2023 Christopher et al., 2024) are therefore inapplicable due the intractable domain of constraints. An alternative approach involves embedding these physical laws directly into the optimization objective as soft constraints, as seen in Physics-Informed Neural Networks (PINNs) (Raissi et al. 2019; 2020). However, this strategy can lead to severe convergence issues without appropriate initial conditions, due to the non-convex nature of these objectives (Krishnapriyan al. [02T]] Wang et al. 2022a).
In response, we propose a Physics-aligned Schrödinger Bridge (PalSB) framework to address these challenges, ensuring both efficient and physically compliant field reconstruction. Our framework integrates a two-stage training strategy that first uses diffusion Schrödinger bridge (DSB) for high-quality field generation from sparse measurements. This initial stage creates super-resolved fields which, while accurate, may not fully comply with physical laws. With this as foundation, the model further refined the second stage through a physics-informed objective tailored to such diffusion bridge, enhancing its adherence to physical principles. Additionally, innovate the sampling process to ensure boundary condition compliance and to streamline generation, targeting efficiency within 10 number of function evaluations (NFEs).
Our main contributions can be summarized as follows:
We explore the application of DSB in physical field reconstruction from sparse measure- ments, focusing on efficient and scalable training strategies that circumvent the need for full field data during initial training phases.
We develop a physics-aligee aligned finettuning approach for generative models to address opti- mization challenges associated with physics-informed loss functions, significantly improving the physical compliance of the generated fields.
We introduce an innovative sampling technique that effectively incorporates boundary conditions into the generative process.
BACKGROUND
PROBLEM SETUPS
Given the low-fidelity measurements denoted as E RT discretized over domain Ω C Rd the goal of field reconstruction to recover the high-fidelity spatiotemporal field X R" over the same domain. The forward mapping from X₂ to y can be represented as y = H(xo) + where H is the observation operator and c is the Gaussian noise introduced by measurement errors. In other words, the measurements can be regarded as a sample of conditional distribution p(y(xo)
N N(y; H((x)), o. "Im)..). To inversely acquire Xo from measurements y, one need to model the posterior probability p(xply) whose form is inaccessible in most scenarios. From either paired dataset N (6) (,,(())))) {0] 1=1 or given prior distribution of xg, such posterior can be statistically approximated (x)(),
data-driven manner.
DIFFUSION SCHRÖDINGER BRIDGE
The Schrödinger bridge problem, stemming from optimal transport and stochastic processes (Schrödinger) 1932 Léonard 2012, Chen et al.) 2016), provides methodological framework to efficiently transition between two probability distributions Pu and P1. This is achieved through the formulation of forward and backward stochastic differential equations (SDEs):
dx, [f(t) + B(tV log (x, t)]dt + / 3(t)dW,
(la)
dx; [f(t) V() log v (x, t)]dt + 3(t)dW,
(1b)
where E [0, 1] is the time variable, {(1) represents the drift term at time ((t) is the diffusion coef- ficient, and dW, and dW, the incremental Wiener processes for the forward and backward SDEs,
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bacoward SD
- -
U-Net f
optimize
8,(,,)
(,(%,)
physical
foward pass +- backpropogation
fineluning
I,(
stop grasiont
Work flow of PalSB. In pretraining stage (top row), the low-fidelity field y is first inter- polated to the same grid for output and the Gaussian-pents -prttrrre linear interpolation of the paired samples is then fed into the neural network to make prediction of the high-fidelity output, where the residual between predition and label utilized optimize the neural network. finetuning stage (bottom row), leveraging the pretrained model, prediction of high-fidelity field is sampled from the low-fidelity condition through the DSB, assessed then by two metrics that evaluate the physical loss and regression loss. Subsequently, the model is tuned through the sampling path using the weighted loss.
Figure I:
respectively. The functions (x+, () and U (x,,44 L) are the forward and backward potentials guiding the transformation between Po and P1- Modifying the drift term f(t) f(t)' 3(t) log (x, t) integrates the formulation into the framework of score-basee generative models (Song et al. 2021). where the score function V log Pt (x, = V log (x, t) (x, f)] guides the evolution of the probabil- ity from data distribution Po a zero-mean Gaussian Pl. Considering that the diffusion process in DSB does not necessarily conclude at Gaussian distribution. additional information can be injected into the final distribution Pt. When the starting and ending distributions are Po(x) and p(y (xo), respectively, the forward SDE models the degradation from high-fidelity to low-fidelity data, which theoretically be reversed via the backward SDE once the function V (x, t) is determined.
Compared to vanilla diffusion models such as DDPM (Ho et al. 2020) and score SDE (Song et al. 2021), DSB is more flexible in alternating the boundary distributions, promoting the direct modeling of inverse problem without starting from pure Gaussian noise. Moreover, the optimality of DSB in transferring distributions
significantly reduce the number of sampling steps. Despite these advantages, solving the Schrödinger bridge problem complex in practice since close-form solution for DSB does not exist in general cases. Some methods approximate the solution using iterative algorithm that needs to simulate the learned SDE (De Bortoli et al. 2021 Chen et al. [022] Tang
al. 2024). while some recent studies introduce the simulation -rree approaches for solving DSB (Tong et al. 2024 Liu et al. 2023b). Among these methods, the 222 approach (Liu et al. 2023b), which simple for implementation, proposes that both potentials conform to the constraints of the probabilistic density function, resulting in corresponding score functions V log V (x,, () and
v log f (xx, for the reversed paths of the described SDEs:
dx, =((()dt + V B(t)dW x $1.00)
(2a)
dx, f(I)dt /B(J)W 1, V(:1)
(2b)
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obs.
interp.
EFE
PIDM
SB
PalSB
ref.
Task (FI)
Task 2 (RI)
Figure 2: Visual comparison between different methods on the two varying tasks. In the first task (FI, top row). the low-fidelity observation (denoted obs.) is 8x down-sampled from high-fidelig field Kolmogorov flow on 256x256 grid. the second task (RI, bottom row), the observation is randomly sampled from high-fidelity field with 99% of the field masked. Our proposed method (SB and PaISB) visually outperforms other baselines that better recovers the spatial patterns as compared the reference (denoted ref.)
Given appropriate boundary distributions, the posterior distributions V I ([xp) and (x,, / become accessible. The posterior distribution is then defined by Nelson's duality (Liu et al. 2023b):
o?
o o
0702 X1: ai 07 + oi
Pr x1) = - Xti
X + (1 +o
(3)
where of = B(T)dT and oj S( ((T)dr represent the accumulated variances. The DSB model is then trained using paired data (xo, x1) in Denoising Diffusion Probabilistic Model (DDPM) style (Ho et al. 2020), with training objective defined as:
I(0) = (x,o(x,,t) X,
X0
(4)
where x, is sampled from the analytic posterior distribution in equation 3
3 METHODOLOGY
The pipeline of PaISB, represented in Fig. [ includes a two-stage (pretraining and finetuning) training process and a particularly designed sampling process.
3.1 DATA-DRIVEN PRETRAINING
In the realm of field reconstruction, the boundary distributions for the DSB in Eq. Tare characterized by the distribution of high-fidelity data, Po- and the distribution of corrupted data, P1. The corrupted data samples are derived from high-fidelity samples through the equation:
xi I(y) (H((x)) +c), Xy Po. o0
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where I denotes predefined interpolation function (e.g., nearest, bi-linear, or bi-cubic) that up- up-scales the sparse observations to match the dimensions of high-fidelity samples.
We find that training directly on high- high-resolution fields dramatically slows the convergence rate and increases the training time for each iteration. To tackle such problem, inspired by the patch- based training widely used in image restoration (Yang al., 2019), we train the model small, randomly cropped local patches from high-resolution field, combined with a scalable network for further inference on global field. This strategy only focuses on capturing the local structure of the physical field but also mitigates the computational burden associated with processing high-resolution inputs. We then reformulate the training objective to directly parameterize the super-resolving operator, as per the following equation:
J(0) E, & ~xropp(xo. Ex spo.Pi [o X1, -xol?
Here, fe represents the field prediction network, defined as fo -0EE + X This network is designed to accommodate inputs of varying resolutions, facilitating scalability to higher-resolution inputs (Luo et al., 2023). The pretraining procedure, including this network's 's deployment, detailed in Alg. [ with further implementation specifics discussed in Appendix A.I
3.2 PHYSICS-ALIGNER FINETUNING (PF)
In contrast to conventional image or video generation, the simulation of physical fields needs to take physical laws into consideration, typically expressed as PDEs. These laws are represented by the constraint F(x) = 0, where FF quantifies deviations from physical laws in sample.
Due the spatiotemporal discretization of the generated sample, perfect conformity with these constraints is challenging to achieve. Otherwise, the most straightforward way is to apply physics- informed losses to the one-step prediction ||F (fo x), as an additional penalty in Eq. 6 However, differing from learning end-to-end model, the score matching objective in Eq. G to an expectation (i.e., not a specific sample) over the Gaussian noise on the DSB, which is a single-step rough prediction of the target field. Accordingly, directly optimizing physies-informed loss on such inaccurate predictions hinders the accurate convergence of the training dynamics.
Instead, drawing inspiration from reinforcement learning with human feedback (RLHF) for diffusion models (Lee et al., 2023) Fan et al.] 2024), our approach seeks to minimize these deviations on the generated samples throughout the generating path by finetuning the model parameters, O. The proposed physics-aligned finetuning (PF) objective is formulated as follows:
(0) = Ey Ex Ex (xoly) [7pakys IF (Xo) || + Treg 1x0 - xoll]
In this equation, Xy represents sample generated along the sampling path, while "phyy and Treg are hyperparameters that balance the loss components, Since the ground truth is available, we introduce an additional regression loss (the second term on the right-hand side of equation (7). which acts as a regularization term. This helps maintain the stability of the optimization process and prevents the model from collapsing into a trivial solution, especially when the underlying constraints are ill posedd This additional term is conceptually similar to the KL-regularization used in DPOK (Fan et al.) (222)) which constrains the tuned distribution to remain close to the pretrained distribution. However, we directly utilize the pretraining samples Xg. whereas DPOK relies on model-generated samples.
Since the whole objective in Eq. 7 is differentiable w.r.t. the model's parameters, we apply a gradient- based method to directly optimize the finetuning objective. The computational graph of the generating process is unrolled to compute the gradients:
OJ, 00
T
OxL. 00
OJ, Oxt.
VoJ,(0)=
(8)
The indices 0 < tr -t-- < < t = represent the diffusion steps used for sample generation,
with T indicating the transpose of the Jacobian matrix. Due to the potentially high memory demand of unrolling the full computational graph, backpropagation is truncated for all steps i T, as suggested
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in the literature Prabbudesaa al. 2023 Clark et al. 2023)
The specifies of this finetuning process
are detailed Alg. and further discussed in Appendix
3.3 SAMPLING STRATEGY
Training focused on local features ensures model scalability and data efficiency but may inadvertently neglect the global coherence necessary for physical field reconstructions, particularly with respect to boundary conditions. Moreover, the inherent scalability of neural networks typically restricts them to learning local mappings, necessitating strategy integrate global features via the learned local mappings. To address these challenges and enhance the speed of the sampling process, we introduce two simple yet effective techniques:
Boundary-aware sampling (BA), Initially, illustrated in Figure 4 boundary-aware (BA) sampling be- gins by padding the low-fidelity input. according to the specific type of boundary conditions. Subsequently, the padded input is interpolated to generate the practical input x used during inference: Notably, this approach is not em- ployed during training because the patch- bashd training methodology in- herently disrupts the original bound- ary conditions.
without BA
with BA
ref.
I (y))
Figure 3: Efficacy of coundary-awand sampling strategy
Early-stop sampling (ES). As ondary enhancement, we find that utilizing smaller step sizes combined with an early stop strategy significantly improves performance compared traditional sampling methods that use larger step sizes for the same NFEs (Fig. 5). Remarkably. in some instances, even single-step generation achieves performance comparable the multi-step generation, as demonstrated in Appendix [A.] and Fig.
4 EXPERIMENTAL SETUP
4.1 DATASETS
Cylinder flow measured by PIV. Under specific circumstance, the fluid flowing around a blund body (e.g., cylinder) can induce periodically falling vortices, which is governed by the Navier-Stoker equations quantifying the conservation laws of mass, momentum and energy. The data are gathered from the particle image velocimetry (PIV) experiment, including only single trajectory of velocity vectors with total length of 879 frames and original resolution of 50 x 67, in which the former 70% of the trajectory is used for training and the rest used for test.
2D forced turbulence. Kolmogorov flow Kolmogorov et al. 1997 Boffetta & Ecke 2012 Chandler
Kerswell, 2013), a canonical system in studying 2D homogeneous isotropic turbulence (HIT) in fluid dynamics, represents more complicated spatiotemporal patterns controlled by the incompressible dataset with spatial resolution of 256x published in (Shu et al. 2023), which contains 40 trajectories with 320 frames in each trajectory. We use 90% of the trajectories as the training data and the rest 10% as the test data.
Navier-Stokes equations. We use the high-fidelit
Reaction-diffusion system. Reaction-diffusion system of the Gray-Scott type (simply denoted as RDGS) is another nonlicar dynamical system widely tackled in many biological (Maini et al. 2004) and chemical applications (Vervloet et al. 2012). The data used here is simulated on 256 256 spatial grid with periodic boundary condition. Two trajectories with 3000 frames that start with different initial conditions used for training and testing, respectively. Notably, with only one trajectory available for training, the amount and diversity of this dataset is much less than the dataset for Kolmogorov flow.

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Detailed descriptions of the datasets and the corresponding physical constraints can be found in A.3 Beyond these 2D cases, PalSB can also work on 3D system. The corresponding description and results can be found in [B.
4.2 TASKS
For each dataset, we train and test the model on two practical tasks in field reconstruction: (1) Fourier- space interpolation (FI (Buzzicotti) 2023)) and (2) real-space interpolation (RI (Buzzicotti) 2023)).
The former is commonly seen in image-bases based measurements such as PIV and BOS (Background Oriented Schlieren method), where the physical field cannot be simultaneously acquired high resolution and covering large field of view, resulting in a great demand for accurate and data-efficient super-resolution method. On the other hand, the latter usually involves in the intrusive measurements such as hot-wire velocimetry and Pitot, which can interact with the physical field itself, leading to restricted number of sensor arrangements.
4.3 BASELINES
We choose the advanced methods in physical field reconstruction basclines, including the conven- tional interpolation method, the End-to-End model based-on FNO (Li et al. 2020) and PIDM (Shu
al. 2023). Detailed description of the baselines be found in Appendix A.2
Interpolation. Conventional approach for filling in missing data for field reconstruction from sparse data. In our experiments, bi-cubic interpolation is used for the Fl task and nearest interpolation is used for the RI task.
End-to-End model with physics-aligned finetuning. Typical style of supervised learning for data- driven reconstruction of field, directly mapping the low-fidelity data to match with the high-fidelity labels. Here, we choose FNO eL al. 2020) the backbone of the neural network, which is suitable for this task (Takamoto et al., 2022). To make fair comparison, we also apply the physics-informed loss function to finetune the end-to-end model.
PIDM (Shu et al. 2023). With denoising network pretrained on high-fideligy data as the prior distribution, DDPM can generate the reconstructed field using proper techniques for injecting posterior information such as SDEdit (Meng et al. (022). In PIDM, by conditioning on the gradient of the equation residual of the input field, physical insatisfaction of generated contents can be implicitly reduced in some cases.
SB. SB method the non-finetuned version PaISB, which is equipped with the same techniques used by PaISB except for the physics-aligned finetuning.
4.4 EVALUATION METRICS
We use two types of metric that can evaluate the point-wise wise accuracy and physical insatisfaction, respectively.
nRMSE refers to the normalized relative mean squared error (Raissi et al. 2020 which evaluate the relative L2 error between the reference field and the predicted field. MSE refers to the mean squared error (I.2 error). MAE refers to the mean absolute error (L.1 error). Correlation refers to the Pearson correlation coefficient. These metrics evaluates the point-wise accuracy of the predicted fields
nER refers to the normalized equation residual (Shu et al. 2023), evaluating the corresponding physical insatisfaction of the given field.
RESULTS
5.1
COMPARISON WITH BASELINES
Figures 2
provides a qualitative comparison between the proposed Physics-aligned Schrödinger Bridge (PalSB) framework and existing baselines, highlighting our framework's 's superior ability to accurately recover spatial patterns in Kolmogorov flow across two distinct tasks; Fl and RI. Additional
comparison between the methods is shown in Figs. 12 15
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Quantitatively, Table displays the metrics used to evaluate the point-wise accuracy and equation satisfaction of the predicted fields from various models. PalSB shows significant improvements over other stateoof-the-l methods across all three physical systems and both tasks. Notably, when finetuned using the physics-informed loss as Eq. 7 the E2E model exhibits higher deviations from the physical laws, particularly in systems governed by highly nonlinear equations such the Kolmogorov flow and the reaction-diffusion system. This can be attributed to challenges in poor initialization of the weights and the tendency of the optimization through a non-convex physics- informed loss to settle into suboptimal local minima, leading to potential convergence failures KKishnapriyan et 2021 Wang et al. 2022a).
Furthermore, the PIDM, while implicitly conditioned on equation residual information, only partially reduces physical dissatisfaction in the Kolmogorov flow and fails other tests. This underscores the limitations of the DDPM framework in scenarios with insufficient data. The spatiotemporally scattered measurements in the RI task can significantly disrupt the dynamics, as evidenced by the nER metric comparison between FI and RI for the interpolation method in Table 2 This disruption hinders reconstruction through SDEdit that starts from the interpolated field, resulting poorer performance of PIDM in the RI task.
Morcover, introducing an additional 5% Gaussian noise to the input low- fidelity data only marginally impacts PaISB's performance, demonstrating its robustness in handling noisy observations (detailed descriptions are available in Appendix B.2).
Table Performance comparison of models across different cases and tasks. The metrics with blue color evaluate the emrors with the reference data, while the metric with red color evaluates the physics compliance.
Case
Task
Model
nRMSE 1
MSR 4
MAB +
Correlation
nER J
Interp.
0.274
2.29E-05
3.06E-03
0.843
5.90E-02
E2E+PF
0.094
3.42E-06
1.08E-03
0.972
8.32F-03
PIDM
0.261
2.14E-05
3.05E-03
0.766
1.60E-02
SB
0.063
1.36E-06
7.18E-04
0.994
1.50E-02
Cy. Flow
PalSB
0.062
1,42E-06
7.04E-04
0.974
1.22E-03
Interp.
0.301
2,80E-05
3,10E-03
0.898
L.70E+00
F2E+PF
0.100
3.81E-06
L.7E.-03
0.906
9,48F-03
PIDM
0.121
4,97E-06
TA4E-03
0.894
4.50E-02
SB
0.092
2,92E-06
1,04E-03
0.972
2.10E-02
PaISB
0.090
2.75E-06
LO2E-03
0.955
8.74E-04
Interp.
0.538
6,58E-01
188E+00
0.857
1.40E-01
E2E+PF
0.288
1,91-011
1.04++00
0,959
3,91E-01
PIDM
0.512
5,95E-01
1.77E+00
0.871
3.45E-01
SB
0.077
1.41E-01
2.24E-01
0.996
7.99E-02
Kol. Flow
PalSB
0.081
1,56E-01
2.66E-01
0.997
2.40E-02
Interp.
0.582
7,69E-01
1.88+400
0.841
6,22E+02
E2E+PF
0.466
5,00E-01
1.72++00
0.908
2.68E+00
PIDM
0.381
3.31E-01
1.34++00
0.934
2.73E+01
SB
0.230
1,23E-01
7.16E-01
0.973
1.95E+00
PaISB
0.253
1.48E+00
8.83E-01
0.973
5.65E-01
Interp.
0.282
2.59E-02
1.0E-01
0.864
3,68E-05
EF
0.261
2.18E-02
1.03--01
0.883
8.17E-05
PIDM
0.268
2,30E-02
1.06E-01
0.870
1.23E-03
0.107
4.25E-03
2.66E-022
0.980
1.58E-06
RDGS
PaISB
0.100
3.76E-03
2.46E-02
0,981
8.91E-08
Interp.
0.292
2.78E-02
9.72E-02
0.855
2,95E-02
E2E+PF
0.394
3,21E-02
1.25E-01
0.883
5.37E-05
RI
PIDM
0.277
1.54E002
9.18E-02
0.918
3,83E-03
SB
0.194
1.34E-02
5.76E-02
0.942
1.86E-06
PaISB
0.193
1.35E-02
5.79E-02
0.941
1.69E-07
5.2
ABLATION STUDIES
Our ablation studies, summarized in Table 2 investigate the effectiveness of specific designed modules within the PaISB framework, with the number of sampling steps fixed at 10. Notably,
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Table Ablation studies across different cases and tasks. The metrics with blue color evaluate the errors with the reference data, while the metric with red color evaluates the physics compliance.
Case
Task
Model
nRMSE
+
MAE
Correlation f
nER
Full model
0.062
1.42E-06
7.04E-04
0.974
1.22E-03
w/o PF
0.063
1.36E-06
7.18E-04
0.994
1.54E-02
FI
w/ DS
0.062
1.42E-06
7.04E-04
0.974
1.22E-03
w/o ES
0.107
3.72E-06
1.23E-03
0.957
2.68E-03
w/o BA
Cy. Flow
Full model
0.090
2.75E-006
1.02F-03
0,955
8.74F-044
w/o PF
0.092
2.92E-06
1.04E-03
0.972
2.05E-02
RI
wo DS
0.092
2.85E-06
1,03E-03
0.944
9.10E-04
w/o ES
0.129
5,50E-06
1,4E-033
0.921
1,98--03
w/o BA
Full model
0.081
1,56E-01
2.66E-01
0.997
2.40F-01
w/o PF
0.077
1.41E-11
2.24E-01
0.997
7.99E-01
F1
w/o DS
0.083
1.64E-01
2.72E-01
0.997
2.57E-01
w/o ES
0.340
2.65F+00
1.10E+00
0.950
3.86F-01
w/o BA 0.109
2.74E-01
3.24E-01
0.994
7.51E-01
Kol. Flow
Full model
0.253
1,48E+00
8.83E-01
0.975
5.65E-01
w/o PF
0.230
1,23E+00
7.16E-01
0.975
1,95F+00
RI
w/o DS
0.259
1.54E+00
9.02E-01
0.974
6.30E-01
w/o ES
0.375
3,21E+00
1.33E+00
0.954
9,08E-01
w/o BA 0.276
1.74E+00
9.51E-01
0.971
1.12E+00
Full model
0.100
3.76E-03
2.46E-02
0.981
8.91E-08
w/o PF
0.107
4.25E-03
2,66E-02
0,980
1.58E-06
FI
w/o DS
0.099
3.74E-03
2.46E-02
0.981
9.51E-08
w/o ES
0.214
1.52E-02
6.64E-02
0.929
9.00E-07
w/o BA 0.103
3.98E-03
2.52F-02
0.980
9.31
RDGS
Full model
0.193
1.35E-02
5.79E-02
0,941
1.69--07
w/o PF
0.194
1.34E-02
5.76E-02
0,942
1.86E-06
RI
w/o DS
0.193
1.33F-02
5.78F-02
0.942
1.90F-07
w/o ES
0.184
1.19E-02
5.62E-02
0.946
7.44E-07
w/o BA
0.196
1.39E-02
5.91E-02
0.940
1.69E-07
physics-aligned finetuning (PF) module significantly reduces the violation of physical laws, even in highly nonlinear and convection-dominat systems, which are typically challenging to optimize from scratch using PINN-based objective (Krishnapriyan et al., 2021] Wang et al. 2022a). This success largely due to data-driven initial point provided by the pretraining stage.
Additionally, several modifications to the sampling process, shown in Table 2 effectively en- hance the final results. These include early-stop (ES), boundary-awar (BA), and deterministic sampling strategies (DS). Specifically, boundary-awan sampling effectively aligns with global effects introduced by boundary conditions, compelling the model to generate content that respects these conditions (see Appendix C.4). Interestingly, removing the Gaussian noise from PaISB's sampling path decreases physical dissatisfaction, and combining this with the carly-stop strategy further im- proves performance. However, removing some modules may slightly enhance point-wise accuracy (nRMSE, MSE, MAE and correlation) while significantly deteriorating performance in terms of nER, indicating severe breach of physical constraints. Further results and discussions these ablations are available in Appendix
RELATED WORK
Data-driven reconstruction of physical field can date from the linear approximation theory such as POD (Berkooz et al. 1993 Everson & Sirovich 1995 Borée 2003 Li et al. 2023). Galerkin transforms NNock & Eckelmann 1994 Boisson & Dubrulle 2011 and stochastic estimation (Adrian) Moin 1988 Suzuki & Hasegawa 2017). in which the performance on complicated systems strongly limited by the linear assumption. Based paired data. further attempts leverage the end-to-end modeling using neural networks, especially, CNN- networks (Fukami et al. 2019
2021a Liu al. 2020) Chai et al. 2020 Ren et al. 2023 and neural operator- based (Li al.

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Published as a conference paper at ICLR 2025
2020).
Drawing inspirations from computer vision in image super-resolution (edigg et al. 2017 Wang et al. 2018: (,22T), adversarial loss and perceptual loss are introduced for field reconstruction
(Yousif et al. | Venkates et al., 2021: Li et al.. 2023; Güemes et al.) (022). These methods
either suffers from poor accuracy on challenging systems (Buzzicotti (002)) nor struggling with
the unstable adversarial training process (Brock al., 2017: 2019 Miyato et al. 2018 [hariwal
& &Nichol [021). More importanly, they can misalign with the corresponding physical constraints.
Encoding of physical laws into field reconstruction are parially investigated in (Jacobsen et al. [023] Shu et al., 2023 Bastek et al. 2024). In particular, instead of directly using PINN loss, PIDM
(Shu et al. 2023) inject physical information through CDM in a classifier-free free guidance manner.
Notably, in many complex nonlinear systems like turbulence, optimizing PINN loss without good initialization is difficult (Krishnapriyan et al., 2021] Wang et al. 2022a) while implicitly informing the physical information like PI-DDPM can easily make the model ignore this extra input.
The literature related to conditional generation of diffusion models and generation on constrained
domain is included in Appendix D
7 DISCUSSION
In this work, we introduce the Physics-aligned Schrödinger Bridge (PalSB) framework, a novel approach for reconstructing physical fields from sparse measurements that effectively addresses the misalignment of physical laws often encountered with diffusion- based models. By employing a patch-based DSB for pretraining, our model achieves robust initial weight configuration. This setup enhances the stability and efficacy of direct optimization when employing physics-informed losses, thereby preventing divergence. The PaISB framework is further augmented by meticulously designed sampling process, enabling the accurate reconstruction of physical fields that adhere closely to physical constraints. Our approach is rigorously tested through practical tasks such as FI and RI, where PaISB demonstrates superior performance compared to baseline models across three different physical systems. These systems include challenging environments governed by highly nonlinear PDEs, such as 2D turbulence and reaction-diie diffusion systems.
The effectiveness of PaISB in these contexts underscores its potential for broad application in generating content that must conform to complex physical constraints within diffusion-based modeling frameworks. The successful implementation of PalSB not only paves the way for more accurate physical field reconstructions but also contributes to the evolving dialogue on integrating physical laws with advanced generative techniques in scientific computing.
8 LIMITATIONS
While the PaISB framework effectively aligns generated physical fields with their governing equations, it is important to acknowledge certain limitations. Firstly, the encoding of constraints within PalSB is implemented in soft manner, implying that some residual discrepancies from the exact equations are inevitable. This soft constraint approach, while facilitating greater flexibility and computational feasibility, does not fully eliminate equation residuals. On the other hand, directly generating complex physical systems, such as those involving turbulence, on the exact solution manifold of the governing equations would be ideal. However, this remains significant challenge due the intricate dynamics and high nonlinearity inherent in such systems. Achieving this level of precision and adherence to the governing equations Moreover, this study is constrained by computational resources, which has limited our validation to two-dimensional (2D) examples. The extension of our framework to three-dimensional (3D) cases, which are more representative of real-world scenarios, has not yet been tested but is critical step for future work. This expansion to 3D will enable a more comprehensive assessment of the model's capabilities and applicability across a broader range of scientific and engineering problems.
generative model an area of ongoing research and development.
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A IMPLEMENTATION DETAILS
A.
PALSB
The algorithms for pretraining, finetuning and sampling of PalSB are included in Algs. 2 and 3 respectively. The interpolation function Z(-), aiming to interpolate from the input space of to the output space Xg. is chosen to be bi-cubic for task while Voronoi tessellation (Fukami et al. 2021b) for RI task.
Compared to training a denoising network that starts from pure noise (such DDPM), the mapping relations for DSB (which start directly from low-fidelity sample) is casier to Iearn. Therefore, the scalable neural network used to predict the high-fidelity field can be a simplified U-Net equipped with
residual blocks and channel-wise linear attention blocks suggested in (Luo al. 2023). removing the group normalization and conventional attention blocks used DDPM (Ho et al. 20200 Song et al. 202T The hyperparameters of the network architecture are listed in Table
sample
pad
I
Figure 4: Boundary-aware sampling strategy
The schematic of BA strategy is demonstrated in Fig. where the input low-fidelity data is first padded according to the boundary condition, and then interpolated to match the dimension of high- resolution output. Subsequently, the sampling process is performed on the padded input, after which the padded part of the generated field is trimmed to obtain the final output. Another trick of ES strategy illustrated in Fig. (5] Under the same NFEs, instead of using a larger step size that covers the full sampling path, stop the sampling procedure the early stage with small step size, regarding the model output from intermediate sample as the final prediction.
sampling steps
1
conventional
step size
etrly-stop
step size
Figure 5: Early-stop sampling strategy
A.2 BASELINES
For the E2E method, we use the FNO (Li al. 2020), which is a powerful tool for learning the mapping between function space, to fit the direct mapping from low-fidelity field to the high-fidelity one. Notably, the low-fidelity input is also interpolated from the original measurements keep the same with method. Since FNO can inherently capture the global interactions within spatial area, we train the model on full fields instead of randomly cropped patches. To make fair comparison, the used for finetuning the PaISB
E2E model is also finetuned using the physics-inforee informed objective
(Eq. 7
The hyperparameters of FNO we used are listed in Table
For the PIDM method, follow all the settings of in the original paper (Shu et al. 2023) that conditioned on equation residual information. Specifically, we use the pretrained weights for the case of Kolmogorov flow published in (Shu et al. 2023), while training the models for cylinder flow and
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Algorithm Pretraining of PalSB
Require:
Ji=o- initialized neural network fo with parameter group 0, diffusion schedule (t). interpolation function I())
Training dataset {xo (f) y f
1: while not convergence do
2: Sample
Get sample from training dataset D Apply random cropping Interpolate
5( <- crop(x) crop(y(1) x()
3:
x + I(((())
4:
5:
Sample t (0,
6:
of l' B(T) of t -: 3(7)dT
7:
Ci नोन. C2 11-
x ~ N ((180
8:
9:
( fo(x)) x
Get prediction step t D. Calculate the loss accroding to eq. P Optimize the neural network
10:
J(0) + |x x01
Optimize (0)
12: end while
8" 8
Algorithm 2 Physics-aligned finetuning of PalSB
Require: Training dataset {xg² (c) (
Ji initialized neural network fo with parameter group 0, pretrained weights 8", physical constraints F(.), diffusion schedule B(t), number of sampling steps T, number of backpropogation step B, weights a,
1: 0 <-0 0"
b Start from the pretrained weights Time scheduling
2: Assign <T < IT-1 <.. <1
3: while not convergence do
4: Sample xu, y (fx (3)
y y()
Get sample from training dataset D Interpolate
5:
XI +
6:
j+l
7:
x X1
8:
while T do
Sampling loop
9:
if j < T
B then
f- sg(x) end if
b Truncate the gradient to save memory
11:
12:
Xo -- fø(x, X1, tj)
13:
at - f dd; o7 +- S1 B(r)dr
a². :i:;co C2 1 C1
14:
C t- अ,
x N C180 C2x1, C107,11
16:
j+ÿ+ +1
end while
18:
T(0) + a|130 Xp ll B((F((
D Calculate the loss according to eq.
Optimize J(0)
20: end while
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Algorithm 3 Sampling process of PalSB
Require: Measurements y, trained neural network fo. diffusion schedule 3(t), number of sampling
steps T, boundary padding function pad(.) and the corresponding boundary trimming function trim(.)
1: Assign < <1 < LT-1 <.. <4= =1
b Time scheduling Pad the field according the boundary condition
2: y pyd(y)
3: X1 I(y)
>> Interpolate
4: x xx
5: j+1
6: while j < T do
Sampling loop with early-stop strategy
7:
No fo(x, so(x,x1,(1) x1
of + le' ((T) o + Sh, B(r)dr
8:
9:
C1 - 3. toj
C2 1-1 1- C₁
x C18y + C2x1
Deterministic sampling path
11:
j j+1
end while
13: 80 < trim(Xn)
Trim the padded boundary grids
ceaction-diffusion system from scratch. Note that the DDPM-based method which start from pure Gaussian noise cannot easily scale to larger domain (Hoogeboom et al. (023). Therefore, we also train the PIDM on full fields instead of randomly cropped patches. All the neural networks for PIDM share the same architecture, where the hyperparameters are listed in the original paper. Based on SDEdit and classifier-fre free guidance, the sampling process of PIDM in our experiments follows the original paper that uses the same parameters as suggested in its Github repository (Shu et al. (202)).
A.3 DATASETS DESCRIPTION
Cylinder flow measured by PIV. Due the limitation of PIV experiment, the pressure field is not accessible, which means the governing equations cannot be fully characterized as physical constraints. Here, we only consider the constraints of continuity (which refer the mass conservatin of the fluid)
as following
11
0
where V. the divergence operator and u is the 2D velocity vector field.
2D forced turbulence. The Kolmogorov flow follows the vorticity equation derived from Navier- Stokes equations written as
dw + u
Vw
22w =f Re
(10)
a
where [u, u] the velocity vector field, w av/Orr du/O is the vorticity field, Reynolds number Re is a constant scalar and f is the external force defined in (Shu et al., (023). Assuming the periodic boundary condition, the velocity in this equation can be easily determined by solving Poisson equation in Fourier space. Therefore, the equation residual can be calculated given a sample of vorticity field.
Reaction-diffusion system. The governing equation of RDGS is written as
Ou
Øt nh, Au - uv + F(1 u)
(11)
Ou
= A.AU + uv - (F + K)Y
Øt
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where and e the concentration variables, Ma and j4, the diffusion coefficients, F and the parameters that control the reaction source terms, and A denotes the Laplacian operator. We use finite difference method to calculate the derivatives in the equations.
A.4 RUNTIME ENVIRONMENT
All the experiments conducted in this paper are running on a single GPU of Nvidia GeForce RTX 3090 with Intel(R) (R) Gold 6226R CPU. The platform Ubuntu 20.04.3 LTS operation system with Python 3.9 environment. We list the parameters for reproducing our experiments in Tables 8 and including the training, finetuning and sampling process.
B FURTHER RESULTS
Table 3: Performance comparison of models on NS3D FI and NS3D RI cases with noise-free and
noisy data.
Case
Model
nRMSE +
MSE 4
L1 Error
Correlation
nER +
PalSB
0.0928
0.0043
0.0457
0.9954
3.8502
NS3D Fl (noise-free)
interp.
0.3263
0.0539
0.1663
0.9415
14.0915
PalSB
0.0948
0.0045
0.0468
0.9953
3,8700
NS3D FI (5% noise)
interp.
0.3267
0.0540
0.1665
0.9414
14.2528
PalSB 0.1846
0.0168
0.0929
0.9820
2,4647
NS3D RI (noise-free)
interp.
0.3645
0.0669
0.1758
0,9299
625.9368
PalSB
0.1850
0.0168
0.0931
0.9820
2.4701
NS3D (5% noise)
interp.
0.3654
0.0672
0.1766
0.9296
629.2205
B. EXTENSION ON 3D DATASET
To validate the capability of our methodology beyond 2D cases, we further test the PaISB on complex 3D system for both FI and RI tasks. The 3D dynamical system is describe by the compressible Navier-Stokes equations, which contains the following continuity equation and momentum equation
0.p+ V (pv) =0 (0.v Vv) - nAv (C+ VV
(12)
where v is the 3D velocity vector, p is the density, p is the pressure, and Ç and nj are bulk and shear viscosity, respectively. while challenging due to the highly nonlinear nature of the governing equations, this problem significant in many applications like aerodynamics.
The training and testing dataset is drawn from PDEBench Takamoto al. (022) with rj Ç periodic boudaries and turbulent initial conditions, which contains 600 trajectories of v, p and Each trajectory contains 21 snapshots spatially discretized on 64 64 x 64 grid. We train our model on the first 90% of the trajectories and test on the rest. We perform 4x super-resolution for Fl task and 1% observation reconstruction for RI task. As illustrated in Figs. 20 and Table 3 PaISB can still make accurate predictions in 3D configurations, representing the power of PalSB in modeling a wide range of physical systems.
B.2 NOISY OBSERVATIONS
In order to assess the robustness of the models, add Gaussian noise with 5% of the standard deviation of the corresponding data into the input low-fidelity field. As shown Table PalSB maintains similar performance
in noise-free cases. Particularly, in the two tasks of reaction- diffusion system where there large area that is not diffused, PalSB recovers these areas even perturbed with noise (Fig. 18 and 19
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FI
RI
Kinetic energy spectral
Kinetic energy speciaal
PusB Inern. EZE-PF
FdeB Hepp
EZE-PF
10
Energy magghtdn
10P
400
200
Eheogy
10
13P
1GF
13F
12F
10
Wave number
Wave number
Figure 6: Statistical comparison of the methods on Kolmogorov flow.
RI
nRMSE
nRR
nRMSE
nER
Cylinder flow
309
Sampling steps
Sampling steps
Sampling steps
Sampling steps
Kolmogorov flow
41/F
Sampling steps
Sampling steps
Sampling steps
Sampling steps
reaclion-diffuiin
7400
Sampling steps
Sampling steps
Sampling steps
Sampling steps
Figure 7: Ablations on sampling steps.
B.3 STATISTICS OF THE KOLMOGOROV FLOW
Kinetic energy spectral (KES) of the turbulent flow (Boffetta Ecke 2012), vital statistics of the energy distribution varying spatial frequency components, is calculated for the fields predicted by different methods as shown in Fig. 6 where the red dot-line represents the KES from the high- fidelity simulation data. In both tasks, the turbulent fields generated by PalSB can better capture the kinetic energy distribution in spectral space than other methods, indicating the efficacy of PalSB in remembering the data statistics.
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C ABLATIONS
C.I NUMBER OF SAMPLING STEPS
As shown in Figs. and 11 though the increase of sampling steps does not necessarily facilitate the accuracy and physical satisfaction, in most cases, more sampling steps can reduce the nER. These phenomena can be first attributed to the restricted sampling steps for finetuning that the model gets overfitted around the configured number of sampling steps. Besides, as investigated in diffusion-based image restoration studies (Luo et al. 2023), the multi-step generation through the diffusion path represent particular capability in generating contents that look like the samples in training dataset, which, in other word, can keep statistical consistency with the target distribution of the dataset while may diverge the point-wise accuracy. Consequently, choose 10 steps to sample the field that makes a proper trade-off between performance and time consumption.
Predictinn errors
Equation residuals
Truncated steps):
Touriated stepta)
0.40
Tou cated step(s)
5
x
7R7/E
0.075
2
5
5
0.30
X
Truncated steptty
Truncated steple)
Truncated stepth
020
steos
steps
Figure 8: Ablations on the number of truncated steps in finetuning, which is tested on FI task for Kolmogorov flow.
C.2 NUMBER OF TRUNCATED STEPS IN FINETUNING
the finetuning stage, the backpropagation along the sampling path truncated to save the computa- tional consumption. The number of truncated steps in finetuning (i.e., B in Alg. 2 can affect the performance of PaISB as seen in Fig. g addition to the higher computational costs, we find that a larger number of backpropagation steps does not necessarily enhance the results. Excessive steps of backpropagation might lead to difficulty for the model's 's convergence to the global optimum within a given number of iterations.
C.3 DIFFUSION SCHRÖDINGER BRIDGE
Based on the observation mentioned above that he increase of sampling steps does not necessarily facilitate the accuracy of the model prediction, we further train the same neural network without using diffusion path, which is, in fact, E2E model that directly optimizes the regression loss. As demonstrated in Ta we find that even such model can achieve a higher accuracy than the model trained by diffusion-like loss, it is still not convincing in characterizing the statistical features of physical field and shows higher violation against the physics. Moreover, such deterministic model is unable evaluate the uncertainty.
C.4 BOUNDARY-
SAMPLING
Considering that the scalability of the neural network used for PaISB results in obstacles for capturing global patterns that introduced by boundary conditions, this strategy is critical for PalSB to enforce the boundary conditions illustrated in Fig. 3. Here, the periodic boundary condition is well aligned through the sampling on the padded field, which enforce the model to be aware the global information in local manner. As shown in Fig. padding is applied to the low-fidelity observation
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Prediction errors
Equation residuals
0.850
padcing
1
padking
podcingg
padbing
0.088
podcingg sizer
0.10
padding stor.
0.85
2
0.80
0.084
0.25
01000
10
30
40
90
10
50
sleus
sieps
Figure 9: Ablations on the size of padding sampling, which is tested on Fl task for Kolmogorov flow.
y before interpolation to the original grid. Consequently, the choice of padding size for Fl and RI tasks depends the size of the observation. Specifically, the padding size along each direction is to 2 for the FI task and 16 for the RI task, respectively. This ensures consistency in the size of the interpolated data samples in each case.
We further examine the impact of padding size on the final performance in the Kolmogorov flow case. As illustrated in Fig. 9 increasing the padding size improves the final performance. However, this improvement comes with additional computational costs, as the neural network must process larger input. Therefore, a trade-off between performance and computational cost should be carefully considered when employing boundary-awan sampling.
without ES
ES
ref.
without ES
ES
ref.
Figure 10:
Ablations etrly-stop sampling strategy. This reaction-diffusion RI task indicates that a lower nRMSE do not mean physically reasonable.
C.5 EARLY-STOP SAMPLING STRATEGY
Under the same number of function evaluation (NFE), we find that sampling with small time step and then stopping the early stage is better strategy than increasing step size as implemented in existing work (Fig. and Fig. 11. Smaller step size can possibly reduce the discretization error for the first few steps on sampling path to more accurately simulate the sample from the posterior distribution intermediate step, yielding significant improvements in most experiments when the NFEs are relatively low (e.g., less than 50 NFEs). Although the nRMSE on reaction-diffusion system shows that the model without early-stop sampling is better, find its failure in generated spatial patterns as illustrated Fig. 10 indicating the bias for only using the point-wiee wise accuracy as the evaluation metric.
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C.6 DETERMINISTIC SAMPLING
Removing all the Gaussian noise in sampling path leads to a deterministic sampling process that is the same as OT-ODE in (Liu et al. 2023b). To our surprise, such procedure shows enhancement on nER in our experiments, even the model is not trained with deterministic DSB (i.e., not trained with OT-ODE). We suppose that this phenomenon related the symmetric noise schedule of DSB, which is nearly noise-free at the two ends of the sampling path, making it possible for the neural network to generalize to noise-free input at the early stage of the sampling process.
Table 4: Performance comparison of models across different cases and tasks noisy observations (5% Gaussian noise). The metrics with blue color evaluate the errors with the reference data, while the metric with red color evaluates the physics compliance.
Case
Task
Model
nRMSE 4
MSE
MAE $
Correlation
nER 4
Interp.
0.276
2.33E-05
3.13E-03
0.842
6.64E-02
E2F+PF
0.094
3.42F-06
1.09F-03
0.972
8.35F-03
FI
PIDM
0.262
2.14E-05
3.0SE-03
0.768
1.59E-02
SB
0.063
1.37E-06
7.22E-04
0.994
1.54E-02
Cy. Flow
PalSB
0.063
1,43E-06
7.10E-04
0.975
1.23E-03
Interp.
0.303
2,85E-05
3,19E-03
0.890
L.73E+00
EFE+PF
0.100
3,82E-06
L17E-03
0.909
9,59E-03
RI
PIDM
0.121
4,98E-06
1,45E-03
0.899
4.59F--02
SB
0.093
2.95E-06
1,0SE-03
0.971
2,04E-02
PalSB
0.091
2.79E-06
1,03E-03
0.953
8.83E-04
Interp. 0.540
6.62E-01
1.88+000
0.856
1.52E-01
E2E+PF
0.288
1.92E-01
1.0E+00
0.959
3.93E-01
FI
PIDM
0.524
6,23E-0T
1.83++00
0.859
3.84E-01
SB
0.085
1.68E-01
2.61E-01
0.996
8.54E-02
Kol. Flow
PalSB
0.088
1.79E-01
2.91E-01
0.996
2.72E-02
Interp. 0.584
7.75E-01
1.86E+00
0.840
6.27E+02
E2E+PF
0.466
5,00E-01
1.72E+00
0.908
2.70E+00
RI
PIDM
0.382
3.33E-01
L34E+00
0.934
2.79E+01
SB
0.233
1.26E-011
7.33E-01
0.972
1,98E+00
PalSB
0.255
1,50E+00
8.90E-01
0.972
5.73E-01
Interp.
0.284
2.61E-02
1.06E-01
0.861
4.63E-04
E2E+PF
0.261
2.18E-02
1.03E-01
0.883
8,18E-05
PIDM
0.268
2.31E-02
1. 1.06E-01
0.870
1.30F-03
SB
0.107
4.27E-03
2.67E-02
0.980
1.59E-06
RDGS
PalSB
0.100
3.77E-03
2.47E-02
0.981
9.13E-08
Interp.
0.294
2.81E-02
1.04E-01
3.02F-02
EFE+PF
0.394
5,02E-02
1.63E-01
0.701
5,39E-05
PIDM
0.2.17
1.SSE-02
9,20E-02
0.918
3,92E-03
SB
0.194
1.34E-02
5.78E-02
0.941
1.87E-06
PalSB
0.194
1,35E-02
5.81E-022
0.939
1.74E-07
D FURTHER DISCUSSION ON RELATED WORK
D. CONDITIONAL GENERATION WITH DIFFUSION MODELS
Diffusion models Song & Ermon 2019 Ho et al. 2020 Song et al 202 are widely applied to
the generation of diverse contents. especially image ge Dhariwall & Nichol 202T Rombach al.
2022.
Saharia et al.) 2022). video (Ho et al., a
a Blattmann et al.) 2023) and molecular (Xu
al. 2022 Watson et 2023 Igashov et al.) 2024). Particularly, conditional generation using diffusion model based given information is highly regarded in practical scenes such as image/video
restoration (Dhariwal Nichol, 2021 Ho & Salimans 2022 Meng et al. 2022 Song et al.
2022
Mardani et al./ 2023 Chung et al. 2023 Song al.
2021 Wang et al. 2022b Yue et al.
2023
Luo et al. [2023] Zhang et al. 2023 Liu et al.] 2023b) molecular generation (Shi et al. 202
Xu
al. 2022 Watson et al. 2023 Igashov et al. 20244
Vecchio et al., 2024 Didi et al. 2024 and
dynamic forecasting (Gao et al.) 2023 Yoon et al. 2023
Cachay et al. 2023), ) basically categorized
into classes that based unconditional diffusion model (UDM) and conditional diffusion model
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Table 5: Ablation on DSB training
Case
Task
Model
nRMSE 4
MSE +
MAE
Correlation ?
nER
w/o DSB
0.063
1.41E-06
7.18F-04
0.973
1.38F-03
FI
w/ DSB (10 steps)
0.062
1,42F-06
7.04E-04
0.974
1.22F-003
Cy. Flow
w/o DSB w/ DSB (10 steps)
0.081
2.35E-06
8.98E-04
0.967
1.44E-03
RI
0.090
2.75E-06
1,02E-03
0.955
8.74E-04
w/o DSB DSB (10 steps)
0.075
1,34E-01
2.30E-01
0.998
3.34E-01
FI
0.081
1.56E-01
2.66E-01
0.997
2.40E-01
Kol. Flow
w/o DSB w/ DSB (10 steps)
0.223
1.14E+00
7.98E-01
0.983
1.0E+400
RI
0.253
1,48++00
8.83E-01
0.975
5.65E-01
w/o DSB w/ DSB (10 steps)
0.092
3.23E-03
2.21E-02
0.984
6.38E-08
FI
0.100
3.76E-03
2.46E-02
0.981
8.91E-08
RDGS
w/o DSB 0.140
6.86F-03
4,18E-02
0.967
1.06F:066
DSB (10 steps)
0.193
1.35R-02
5,79F-02
0.941
1.69F-077
(CDM) respectively. Utilizing a pretrained UDM as priori distribution, methods such as (Ho & Salimans 2022; Chung al. 20233 Song et al. 2022, Wang al. 2202bb Mardani et al. 2023
manipulate the generating path towards the required condition, which is usually performed through
the gradient-based correction. By partially noising the conditions along the forward diffusion path and then running the reversed process, SDEdit (Meng et al. (022)) can approximately sample from the target conditional distribution. ControlNet another popular tool appended to a pretrained UDM, injecting conditional information through finetuning additional feature adjusting network on paired data. Classifier- ffreeerrr-feeeeee guidance (Ho & Salimans 2022) uses a weighted combination of UDM and CDM to guide the sampling path. Instead of training UDM that requires intensive data and computational resources, CDM directly add the conditional information as an additional input the model. As a special case of CDM, instead of starting from Gaussian noise, DSB (De Bortoli et al. [021] Bunne ett al.] 2023] Liu et al. [023b] Tong et al. 2024) Yue et al.) 2023) directly learns the probabilistic path from the conditions the targets, which is much more fast-to-train trii and data-efficient for specific task. Despite the empirically proved efficacy on image/audio tasks, the modeling of physical field
using DSB is not well studied, which is addressed in this work.
D.2 GENERATION ON CONSTRAINED DOMAIN
Diffusion model is adept to characterize the statistics of training data yet does not guarantee the satisfaction of constraints for the generated contents. There are efforts to apply simple constraints on diffusion models. Specifically, pixel-wise thresholding during generation (Ho al. 2020 Saharia ct al. 2022) Lu et al., 2023) is a simple while effective trick to fulfill the value range of an image.
Similarly, mirror diffusion model (Liu al. 2023a Lou & Ermon 2023) leverages the mirror mapping to restrict the generation not exceeding a given convex set. Generation with equaivariance (Hoogeboom et al. 2022 Igashov et al., 2024) is another class of methods that are ubiqitous in molecular related problems. Projected generative diffusson model (Christopher et al. (0244) further extended the constraints to more complicated domain on the support of IIGDM (Song et al. 2022). However, these methods can only work on tractable domain of constraints, failing when the sample is embedded on a complicated and intractable manifold such as the physical field that derived from the solution of highly nonlinear PDEs.
E CODE AND DATA AVAILABILITY
The code for PaISB is submitted as the supplementary material. The data and trained checkpoints can be downloaded at the Google Drive after the paper's acceptance.
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FI
RI
nRMSE
nRR
nRMSE
nRR
Cylinder flow
Sampling steps
Sampling steps
Sampling steps
Sampling steps
Kolmogorov flow
Sampling steps
Sampling staps
Sampling steps
Sampling steps
reaclion- reaclion-ciffusion din-diiffssinn
15
Sampling steps
Sampling steps
Sampling steps
Sampling steps
Figure 11: Ablations on sampling steps (5% noise).
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obs.
interp.
E2E+PF
PIDM
SB
PalSB
ref.
10
Figure 12: Additional results of Kolmogorov flow on Fl task (8x super-resolution, noise-free).
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obs.
interp.
E2E+PF
PIDM
SB
PalSB
ref.
10
-10
Figure 13: Additional results of Kolmogorov flow on RI task (99% masked, noise-free).
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obs.
interp.
E2E+PF
PIDM
SB
PalSB
ref.
Test
1.0
M
0.5
MB
1.0
0.3
0.5
0.0
Figure 14: Additional results of GS-RD on Fl task using noise-free observations (8x super-resolution, 0% Gaussian noise).
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obs.
interp.
E2E+PF
PIDM
SB
PalSB
ref.
1.0
M
0.5
1.0
0.3
0.5
0.0
Figure 15: Additional results of GS-RD on RI task using noise-free observations (99% masked, noise-free).
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obs.
interp.
E2E+PF
PIDM
SB
PalSB
ref.
-10
Figure 16: Additional results of Kolmogorov flow Fl task using noisy observations (8x super- resolution, 5% Gaussian noise).
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obs.
interp.
E2E+PF
PIDM
SB
PalSB
ref.
-10
Figure 17: Additional results of Kolmogorov flow on RI task using noisy observations (99% masked, 5% Gaussian noise).
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obs.
interp.
E2E+PF
PIDM
SB
PaISB
ref.
0.3
0.5
0.0
1.0
0.3
0.5
1.0
0.3
0.5
0.0
Figure 18: Additional results of GS-RD on Fl task using noisy observations (8x super-resolution, 5%
Gaussian noise).
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obs.
interp.
E2E+PF
PIDM
SB
PaISB
ref.
0.3
0.5
0.0
1.0
0.3
0.5
1.0
0.3
0.5
0.0
Figure 19: Additional results of GS-RD on RI task using noisy observations (99% masked, 5%
Gaussian noise).
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density
pressure
obs.
Interp.
PaiSB
ref.
Figure 20: Extension on 3D case for FI task (4x super-resolution, noise-free).
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density
pressure
obs.
Laxt
Interp.
PaiSB
Figure 21: Extension on 3D case for RI task (99% masked, noise-free).
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Table 6: Parameters of the neural network used in PalSB
Parameter name
Cylinder flow
Kolmogorov flow
Reaction-diffusion
Model architecture
Denoising Luo et al. (023)
DenoisingUNet
DenoisingUNet
Bunin
0.1
0.1
fmax
0.3
0.3
0.3
Number scales
1000
1000
1000
Positional embeds
sinusoidal
sinusoidal
sinusoidal
Number features
32
Channel multiplier
(1, 2, 4, 8) 2
(1, 2, 4. 8, 16)
(1, 2, 4, 8, 16)
Number residual blocks
Nonlinearity
Swish
Swish 0.1
Swish
Dropout
0.1
0.1
Table 7: Parameters of FNO
Parameter name
Cylinder flow
Kolmogorov flow
Reaction-diffusion
Model architecture
2D FNO
Li et al. (2020)
2D FNO
2D FNO
Modes
16
16
16
Modes 2
16
16
16
Number of features
32
32
32
Number of blocks
4
4
4
Nonlinearity
GeLU
GeLU
GeLU
Table 8: Parameters for training
Parameter name
Pretraining
Finetuning
Optimizer
AdamW
AdamW
Learning rate
lc-4
le-5
Learning rate step
1000
10
Learning rate decay
0.99
0.99
Batch
64
32
Small batch size Number of iterations
64
4
100000
1000
Ema rate
0.995
Sampling steps T
10
Sampling step size
1e-3
Backpropogation steps
Table 9: Loss weights for finetuning
Parameter name
Cylinder flow
Kolmogorov flow
Reaction-diffusion
Iphys
5
0.5
le5
Treg
I
10
37


Reviews:
[{"review_id": "ZQrBVtQ8lF", "reviewer": "Reviewer_BWVW", "summary": "In _Physics-Aligned Field Reconstruction with Diffusion Bridge_, the authors propose a method for reconstructing physical fields from a limited number of measurements. They combine a diffusion Schrödinger bridge, following the $I^2SB$ approach by Liu et al., with a fine-tuning step to improve consistency with physical constraints. Additionally, they adapt their sampling procedure to be \"boundary-aware\" by padding the input appropriately and achieve good performance with a small number of function evaluations through \"early-stop sampling.\"", "questions": "Questions / Suggestions for Clarification:\n1. In line 083-085: What is the relationship between $m$, $n$ and $d$?\n2. I can't quite follow the derivation of eq. (6) from eq. (4), I end up with $f_\\theta + x_0$ instead\n3. line 268: Doesn't the backpropagation get truncated for all steps $i<N$? That's how I interpret both Fig. 1 and Alg. 2\n4. Does finetuning use early stop sampling?\n5. Does truncating the backpropagation in the finetuning step affect the adherence to physical constraints in a time-step dependent manner? I think this is alluded to in Appendix C1 but I think it warrants further clarification, especially any potential interaction with early stop sampling.\n6. How much padding is used for sampling and does the amount of padding influence performance? Does that interact with the number of sampling steps?\n7. Algorithms 2 and 3 are not totally clean:\n   - 7: while $j=T$ -> while $j<T$\n   - 11: $t_j$ not defined\n   - 12: $t$ should instead be $t_j$\n\nTypos/Format etc\n- line 065: such diffusion bridge -> such a diffusion bridge\n- missing year in reference Fan et al.\n- line 329: should this say the physical field cannot be simultaneously acquired at high resolution and covering a large field of view? Surely the resolution and FOV are known.\n- line 883: calculate the loss according to eq 4 -> calculate the loss according to eq 6\n- line 1063: \"observation mentioned above\" -> vague, which one?\n- Fig 10+11: noisy -> noise-free", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The paper addresses the important challenge of reconstructing physical fields, drawing inspiration from recent methods such as $I^2SB$ diffusion Schrödinger bridges and reinforcement learning with human feedback for diffusion models. The authors convincingly demonstrate the effectiveness of their method by evaluating it across three datasets (with two tasks each) and comparing it against several reasonable baselines. Further, they conduct an ablation study, reinforcing the value of their design choices. The early stop sampling strategy in particular could prove to be useful outside of the context of physical field reconstruction as well. The method and results are presented clearly with only minor clarifications needed.", "weaknesses": "There's no major weaknesses of the paper.\nFor minor opportunities for clarification and cleanup see Questions.", "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": "S6jrnqqH9e", "reviewer": "Reviewer_bXrf", "summary": "The paper presents a method for reconstruction of physical fields from sparse information using Schrodinger bridges. The setup uses a dual training approach where the diffusion bridge is first trained to bridge between sparse and high resolution measurements, then it is fine tuned to satisfy physical constraints. This is coupled with a special technique to care for boundary conditions. The method is tested on several examples of physical field reconstruction tasks.", "questions": "Can you provide strong counterpoints to the above weaknesses?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- physical field reconstruction is an important task with potential high impact in applied sciences\n- the presented method is well-chosen for the problem\n- the method is experimentally validated on several example problems", "weaknesses": "- while the methodology is well chosen, I don't believe the paper presents major new methodological contributions\n- some of the techniques seems somewhat adhoc, i.e. employing early stop sampling. I don't doubt that these techniques work well for the problem at hand, but they are not interesting as such from a methodological point of view\n- the experimental validation is somewhat limited being restricted to 2d problems and with fairly limited datasets. A more extensive validation that clearly demonstrates the power of the method would make a stronger case for the paper, given the limited methodological contributions", "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": "muPuq7jX9Z", "reviewer": "Reviewer_bWQy", "summary": "This paper presents the Physics-aligned Schrödinger Bridge (PalSB) method, developed for reconstructing physical fields from sparse data subject to physical constraints. Reconstruction of physical fields is a key task in fields such as fluid mechanics, meteorology, and astrophysics, where spatiotemporal information about systems must be recovered from sparse, often noisy, observations. Despite significant advances in machine learning and the use of diffusion models to solve complex data problems, many existing approaches fail to satisfy physical laws and struggle to cope with the nonlinear nature of complex physical systems.\n\nPalSB proposes a new approach to data reconstruction that incorporates a diffusion bridge mechanism (Schrödinger Bridge) specifically adapted to satisfy physical constraints. The model is trained in two stages: the first stage focuses on local reconstruction map construction, and the second stage focuses on global physical principles. This combination allows for the creation of super-resolved data that not only more accurately but also better satisfy physical constraints than similar methods.\n\nThe pretraining stage of PalSB is based on a diffusion Schrödinger bridge, which allows for efficient transition between high-quality and corrupted data distributions. To reduce computational costs and improve the accuracy of the model, the training process is performed on small local patches of the high-resolution field. This helps the model focus on the local structure of the data, avoiding the need for large computational resources. The second stage involves Physics-aligned Finetuning, which allows the model to fit physical laws represented as differential equations. This stage minimizes deviations from physical laws using a combination of physically informed and regression loss functions.\n\nThe paper also proposes boundary-aware sampling to handle boundary conditions and early stopping sampling, which improves the generation efficiency and reduces the number of required steps. These strategies allow the model to better account for global conditions and significantly improve the performance of PalSB.\n\nThree complex physical systems were chosen to evaluate PalSB: flow around a cylinder, two-dimensional turbulence, and a reaction-diffusion system. Based on the benchmark results, PalSB demonstrated significant improvements in accuracy and physical compliance compared to other methods, including interpolation, end-to-end models, and physically informed diffusion models (PIDM). In particular, PalSB significantly reduces physical inconsistencies and improves the compliance with spatial patterns in both Fourier and real space interpolation problems.\n\nThe study also showed that removing individual PalSB modules (e.g., physically consistent tuning or boundary-aware sampling) degrades the physical compliance and model performance. This highlights the importance of PalSB's integrated approach to achieve high accuracy and compliance with physical laws.\n\nIn conclusion, PalSB demonstrates great potential for field reconstruction in complex systems, showing high quality data recovery and improved compliance with physical constraints. In the future, PalSB is expected to be extended to 3D problems, which will cover a wider range of scientific and engineering applications that require compliance with physical laws during the generation process.", "questions": "Can you additional explain me, why did you use only normalized metrics? Also, can you please explain why you checked on flow equations (like Kolmogorov and Reaction-Diffusion equations)? I highly recommend checking on electrostatic equations as well.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "First, it provides high accuracy, surpassing traditional methods by using the diffusion bridge mechanism, which allows for efficient data recovery from sparse measurements. Second, PalSB integrates physical constraints (boundary conditions and equations) at the fine-tuning stage, which minimizes mismatches with real physical laws.", "weaknesses": "Unfortunately, I do not understand your motivation for using only the normalized error. I highly recommend that model errors can be estimated using the mean squared error (MSE) and L1 error metrics. In addition, the residual error, which determines how well the predicted field fits the equations describing the physical system, can be used to assess compliance with physical constraints. Also, for the future, I highly recommend combining your approach and this approach https://arxiv.org/pdf/2403.14404", "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 (Spotlight)