Partial Physics Informed Diffusion Model for Ocean Chlorophyll Concentration Reconstruction
Authors: Qianxun Xu
Zuchuan Li
Venue: NeurIPS 2025 poster ()
Keywords: Diffusion Models
Physics-Informed Machine Learning
Ocean Data Reconstruction
Primary area: machine_learning_for_sciences
TL;DR: Generation of physically plausible results when only observations and partial physics knowledge of the system is available.

Abstract: The integration of big data, physical laws, and machine learning algorithms has shown potential to improve the estimation and understanding of complex real-world systems. However, effectively incorporating physical laws with uncertainties into machine learning algorithms remains understudied. In this work, we bridge this gap by developing the Partial Physics Informed Diffusion Model (PPIDM), a novel framework that integrates known physical principles through a physics operator while reducing the impact of unknown dynamics by minimizing related discrepancies. We showcase PPIDM’s capabilities using ocean surface chlorophyll concentration data, which are influenced by both physical and biological processes, while the latter is poorly constrained. Experimental results reveal that PPIDM achieves substantially improved prediction accuracy and stability, significantly outperforming baseline methods that either neglect physics entirely or impose incomplete physical constraints under the assumption of completeness.

Partial Physies Informed Diffusion Model for Ocean Chlorophyll Concentration Reconstruction
Qianxun Xu
Zuchuan Li
Division of Natural and Applied Sciences
Division of Natural and Applied Sciences
Duke Kunshan University
Duke Kunshan University
Kunshan, China
Kunshan, China
qianxun Xtanxunnxx@dukeele xudukkkkkkkkukkeennnnnnnaaaa edu. en
zuchuan hittmmmm.idduukekusshhnnn edu. cn
Abstract
The integration of big data, physical laws, and machine learning algorithms has shown potential to improve the estimation and understanding of complex real- world systems. However, effectively incorporating physical laws with uncertainties into machine learning algorithms remains understudied. In this work, we bridge this gap by developing the Partial Physics Informed Diffusion Model (PPIDM), a novel framework that integrates known physical principles through a physics operator while reducing the impact of unknown dynamics by minimizing related discrepancies. We showcase PPIDM's capabilities using ocean surface chlorophyll concentration data, which are influenced by both physical and biological processes, while the latter is poorly constrained. Experimental results reveal that PPIDM achieves substantially improved prediction accuracy and stability, significantly outperforming baseline methods that either neglect physics entirely or impose incomplete physical constraints under the assumption of completeness. Code will be available here
Introduction
Diffusion models generate samples from an unknown data distribution by reversing a forward noising process applied clean data. They have demonstrated remarkable success in generating complex textures, structures, and motion patterns across a wide range of applications, excelling in generative tasks such as image synthesis
video generation 6 9
and medical analysis (20] 22
Despite these advances in generating content that is coherent and closely aligned with the under- lying data distribution, diffusion models still face challenges when applied scenarios where the generated data must strictly adhere to specific constraints. This is particularly evident in scientific and engineering applications, where the generated data must not only mimic real-world examples but also meet strict specifications and comply with fundamental physical laws. However, training diffusion model a dataset that meets specific constraints does inherently ensure that the generated samples will strictly conform to those same constraints. As result, incorporating explicit domain knowledge essential for guiding the model toward more sophisticated understanding of the data distribution and its underlying physical principles. Recent studies I [5] 1II 17 [8]
have made notable progress in this regard by embedding physical constraints directly into the models. These approaches typically assume that data can be fully characterized by well well-defined constraints. In practice, however, these constraints are frequently incomplete, either due the absence critical parameter values or because of limited understanding of underlying processes. Consequently, existing methods such as minimizing the residual to zero 17 or constraining the generated data within predefined bounds 2 may not guarantee physically consistent generations.
"Corresponding author.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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To address these challenges, we propose the Partial Physics Informed Diffusion Model (PPIDM). which captures the reliable portions of the governing laws through a physies operator yet remains flexible to unmodeled or uncertain system components. PPIDM applies this operator to both real and generated data and penalizes discrepancies between their outputs, while allowing unmodeled or uncertain components of the system to be learned from data. This approach bridges the gap between theoretical constraints and data-driven adaptability, resulting in more accurate and physically consistent generations without compromising the model's ability to capture unknown partially understood dynamics.
We demonstrate the performance of PPIDM on the reconstruction of oceanic chlorophyll (Chl) concentration, a task that involves infilling temporal gaps and predicting future values. Chl is widely used proxy for phytoplankton biomass and oceanic primary productivity. Accurate reconstruction of Chi is crucial for understanding oceanic biogeochemical processes, monitoring ecosystem health, and assessing the ocean's response to climate change. Chl dynamics is governed by both physical and biological processes, and is often modeled using the equation (ADR PDE). The advection term of ADR can be well constrained using velocity fields, while the diffusion and reaction terms are more difficult to observe. In particular, the reaction term, encapsulating biological processes such as phytoplankton growth, depends on complex factors such as nutrient availability, light conditions, and community composition which are often intractable. Therefore, reconstructing models solely constrained by advection are incomplete and inaccurate. To address this, PPIDM integrates partial physical knowledge with data-driven learning. Our experimen- tal results ocean Chl data (2n] demonstrate that PPIDM effectively balances domain knowledge with observational data, outperforming purely data-driven baselines and naive implementations that assume complete physics.
Reaction partial differential
Related Works
2.1 Denoising Diffusion Probabilistic Models
Diffusion models are class of probabilistic generative models that learn to map samples from the true data distribution (<)) into pure noise via a forward noising process, and then learn to invert this process to recover data from noise using a learned model distribution po(
20).
Specifically, the forward diffusion process introduces Gaussian noise progressively to an initial data point Xo through a Markov chain across T discrete steps. At each timestep 7, noise is injected according to:
q(x xx-1) = A(xx; v 1,,,
where the noise level is controlled by pre-defined variance schedule 3r E [0, 1]. Due to the Gaussian nature of each incremental step, the marginal distribution q(x, xo) any timestep can be derived in closed form:
q(>i xx) = N(x: v /a (1 - I))
(2)
where a = 1 - Br and a. = II. as.. As + T, the distribution converges toward a standard Gaussian NO I), making Xy essentially pure noise.
The core challenge for diffusion models lies in reversing this noising process to generate samples from the data distribution gXx)). Ideally, we would sample directly from the true posterior distributions q(x,-1 xt). However, since these distributions are analytically intractable, diffusion models approximate them using parameterized conditional distributions (x x.) modeled by neural networks:
po(xor)) p(x))
II (x xx), (=1
where each conditional is defined as a Gaussian distribution parameterized by learned functions oo [x, t) and Eg (xx, t). Typically, p(xr) R. (0, I), enabling iterative denoising from noise to the original data. Training maximizes a variational bound on log pogp), under the usual formulation this reduces to the noise prediction objective 77:
L =Exo.ei [|ee Ex(x((xoxE 11?
2

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where E E,,,) This loss is equivalent to predicting the clean sample,
X
X - /1 0t (xxx,,)))
(5)
In this work, we explicitly frame our objective in terms of predicting the clean signal Xg. because this allows for more straightforward integration of physical constraints on the reconstructed state.
2.2 Physies Informed Machine Learning
Diffusion models have recently been extended to incorporate physical knowledge for scientific modeling. One prominent line of work enforces governing equations directly during training or sampling. For example, CoCoGen [11] incorporates discretized PDE constraints into the reverse diffusion process to ensure physically plausible generation. Similarly, PIDM T introduce PDE residual losses into the training objective align generated samples with known physical laws. These models offer high fidelity under well-specified physics, but assume the governing equations are both complete and accurate, which often breaks down in real-world systems. Other methods condition the diffusion process on physics- physics-derived derived signals. Projected Diffusion projects the state at every step onto constraint consistent manifolds that encode physical feasibility [2] whereas DiffusionPDE learns joint distribution over coefficients and solutions and performs inference with sparse observations and physies guided updates [TO]. Though effective, these models rely on fully known governing equations to define constraints or training distributions, and are less suited to settings with incomplete knowledge.
Beyond diffusion--ar based approaches, physics-informed machine learning frameworks 13 15
16 17 address partial knowledge by estimating unknown parameters learning solution operators, then simulating forward from initial or boundary conditions. Although these methods accommodate inverse settings and incomplete physics, their primary objective PDE identification or operator learning, and they typically yield a single forward trajectory per initial state. Incorporating irregular, multi-time conditioning inference such as conditioning on arbitrary subsets of observed frames and generating ensembles of reconstructions typically requires additional data assimilation or explicit stochastic modeling beyond the base frameworks. This requirement limits the flexibility of these methods in real-world scenarios characterized by irregular observations.
In this work, study diffusion models in which the governing physics is only partially known. Instead of requiring full equations or simulators, PPIDM introduces a physics operator ó that encodes only the trusted components of the dynamics and couples this partial theory with data-driven denoising through physics residual difference. This design extends physics-informed diffusion to under-sped seeei--peeified scientific systems and leverages the generative nature of diffusion condition on any subset of observations, enabling the generation of multiple physically plausible reconstructions without retraining.
Partial Physics Informed Diffusion Model (PPIDM)
3.1 Problem Formulation and Physies Operator Construction
We consider a physical system whose true dynamics stem from both known physics and unobserved biological processes. Concretely, the state variable x(l) evolves according to:
Ox
P Panowa (x, v) + Bunknown (x)imnoww(x)))
(6)
Ø1
where Pknown models known processes such as advection with velocity field v, and Bunknown en- capsulates latent biological effects other unresolved physics. Our main goal is reconstruct the Chl concentration state while preserving consistency with Pknown- The core challenge lies in enforeing partial physical knowledge without over-constraining the model where full dynamics remain unknown. Therefore, we define a physics-informed operator:
: X' +
where denotes the space of original states (e.g., concentration fields or other physical quantities to be generated) and y represents the space of physics-ifformed projections. This operator ó is
3

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Physics-Informed Subspace
Data Space X
Physics Operator (x)
Updated model weights
Partially known physics information
0 - 0 1/Vlllxx xall² + R(A0..)) c)
CD
Estimate E[xo|x+]
X
x
x
Denoising step with Physics Alignment
Xg
Figure 1: Overview of our proposed PPIDM: At each denoising step, the model projects predicted and ground truth clean signal onto the physics-informed subspace and updates model weights with auxiliary physics loss term, guiding the model to learn the partially known physics.
constructed so that applying & to a ground-truth or predicted states enforce and reveal consistency with the known physical laws. Specifically, for any state Xg. 6(xx) can be viewed as its projection into a physics-iforrmed subspace, which is the set states that are consistent with the known but incomplete dynamies, or how the system would look if only the known physics governed it. The exact form depends on the level and nature of the available physics knowledge:
When the governing equation is known in principle but contains unknown parameters, we selectively remove or omit terms involving those unknowns. We then build by applying the remaining or known part of the equation.
Partially Known Parameters:
Fully Known Subsystem: When law perfectly describes partial subsystem and all associated parameters are certain, ø defined to enforce this subsystem exactly. Even though the law itself is fully accurate for its domain, it does address the rest of the system's dynamics. By applying ø to each sample, we ensure that known subsystems are satisfied, leaving the unknown effects such as additional more complex processes be learned from data.
3.2 Mechanism of the Physics Operator
Let X denote ground truth sample from the data distribution and &o the corresponding model- generated sample at each step t. In a vanilla diffusion model, one typically minimizes loss directly between Xy and kn:
Caata = Exn.ett [w(t)]lxo- (xx,x)t))
(7)
where w=() I-as is the weighting function and So is the estimate of original data at each I-a. time step of reverse process utilizing Eq. 5 However, this loss is purely data-driven and does not incorporate domain knowledge. To leverage the physics operator Φi unlike prior work that enforces Φ( ((x)) 0 based on fully known physical laws [1]. we recognize that our system evolves under both Pknown and Bunknown- As such, the true state X9 itself does not strictly satisfy d(xo) = O, so unreasonable to force the predicted state to satisfy known physics absolutely. Instead, we design the physics loss to encourage Xo to satisfy the known physics the same extent as the true state xo does. Formally, define physics residual difference that quantifies the discrepancy between the projection of the predicted state and that the ground truth:
AΦ = Et (x(x)) - Φ (xxx (x.,
(8)
To avoid over-constraining the model at early timesteps when the predicted states are still highly uncertain, adopt progressively enforced constraint following the probabilistic formulation of Q.

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Specifically, we interpret the residual difference Aø as a realization from a zero-mean Gaussian distribution with timestep-dependers variance Σ, yielding the following loss:
Sphysics = R(AØ, t) A - log [A 0, 2,)
where À is a scalar coefficient that controls the contribution of the physies term. In practice, M, is obtained directly from the diffusion model's posterior variance schedule at timestep t, which naturally reflects the model's 's uncertainty over the denoising trajectory. As the reverse process proceeds and the posterior variance decreases, the likelihood sharpens, increasingly penalizing deviations from physical constraints. This construction induces an implicit deneddeenddnt weighting. Early in the process, when uncertainty is high, the model prioritizes recovering the coarse structure of the data and allows flexibility the residual. Later, as predictions become more confident, the loss enforees stronger adherence to physical consistency.
This formulation is particularly important for systems that combine well-understood components with poorly constrained processes. Our Eq, exxplicily integrates the known physical dynamics captured Eq. 6 while treating the unknown uncertain components as noise.
The training loss of our algorithm is then a combination of the data fidelity and physics residual term represented follows:
Luvtal = Liaaa + Lphysies:
(10)
At inference time, we aim to reconstruct the clean state Xo from noisy initial sample xT using the standard reverse diffusion process. Notably, while our model incorporates partial physical knowledge during training through the physics projection operator ø, this operator is not applied during inference. This is because external physical inputs are typically unavailable for frames or timesteps that do not exist yet (e.g., in prediction or infilling tasks). We generate the unknown frames by sampling from the learned reverse process while keeping the known frames fixed, following a standard clamping strategy used in conditional diffusion models (9)
4 Experiments
4.1 Data Preparation and Training
We train our PPIDM on the Chi concentration and velocity field data from the Biogeochemical Southern Ocean State Estimate dataset (21). The dataset spans 2008 to 2012, with spatial resolution of 1/6" and temporal resolution of days. The data consists of time, latitude, longitude and other attributes, and can be visualized temporal continuous images in the Southern Ocean. Given the approximately log-normal distribution of Chi measurements [12, we apply a logarithmic transform during preprocessing. We gather the Chl data along with the corresponding velocity field data on the horizontal directions u and a of the ocean surface at each timestep. We segment each image into 64 x 64 patches with sliding window, discarding any tiles containing landmass. This cropping expands the number of training samples while maintaining the essential oceanic regions. During sampling. the complete image given timestep can be reconstructed by independently sampling each image patch and assembling them into a full frame.
Additionally, we organize the training data by slicing across the time dimension, allowing the model to learn temporal dependencies. Specifically, for each 64 x 64 region, we build training sequences (Xx,, X 1: ......-.TTTIll 1), (Xn+1, Xn+2, ..., Xn+T),...h where xx, represents the Chl state at time index nı, and T' represent the length of the window for training. Sliding windows that extend beyond the available temporal boundaries are discarded. The test set is designed to be the first T timesteps of each spatial region, with a 2T timestep buffer following these test frames excluded from training to ensure independence of spatial patterns between the sequences used in training and testing. Notably, inference can be performed on a sequence of any lengths. For consistency, we set T = 20 for training single NVIDIA RTX 4090 GPU. We set
and inference in all experiments. We train all models the number of data loading workers to 16, the batch size to 64, and the learning rate to 1 x 10-4.
4.2 Experiment Setup
We consider two representative scenarios to evaluate the model's ability to integrate partial physical knowledge, following section 3.T These scenarios reflect common situations in many research areas, including occanography, where only certain aspects of the underlying dynamics are known.
5

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10
11
12
18
17
15
10
piel
pldl
pdet
pddd.31
Trxe Typoe
lnch1
lock!
1arkn.1
Terkh
lach0.31
back0.1
3.06
0.30
0.12
0.05
0.30
014
30SE
Values
(a) Region A reconstruction MSE
(b) Region B reconstruction MSE
The reconstruction MSE with different weights of physics operator for different regions. Loss types have format <op><>>, where op (pde, back) and {0.001, 0.01, 0.1, 1] (pde: ADR PDE physics operator; back: backtrack physics operator). Given frames and 20, reconstruct frames 2-19, and repeatedly sample for 20 times form the box plot. Different regions favor different operator-weight pairs, reflecting inter-region variability in physics dominance.
Figure 2:
Case I: Partially Known ADR Parameters. the ADR PDE:
Chi dynamics at the ocean surface is modeled using
0x(() + V. (u(()x(()) = V (D(()) Vx()) + R(I),
(11)
Øt
where the advection term is well specified, but the diffusion D(x. t) and reaction R(-) terms remain poorly constrained. We train our model with a constraint derived from a physics operator ø consisting of the local change rate and advection terms (i.e., material derivative). This operator is applied the Sg as follows:
0xo(t) Ot
(((())x(())
((Ro)
(12)
An analogous transformation ((x;) is computed for the ground-truth field x Minimizing the discrepancy between of (xxx) and Φ (8)) encourages the model to learn predictions consistent with the known advection portion of ADR PDE, leaving diffusion and reaction terms to be inferred from data. We denote this as the pde physics operator.
Case 2: Fully Known Particle-Trackee Law. next consider a situation where the advection of Chl parcels itself is precisely known. We can track the positions of existing parcels backward in time based on measured horizontal velocities, but parcel emergence and disappearance due to biological processes remain untractable. Specifically, for each point (æ, y), velocities (u, v). and small increments At, Ax, Ay, the law for backtracking particle positions is:
Dback = I Udata (x, y, t) At Ar,
(13)
Dback Vaata (z. y. t) At Ay,
where Pbacks Mback denotes the backtracked position of parcels time t A.. To formalize this known transport mechanism, we define physics operator & that maps any field x(x, y, to its backtracked value:
d(((x, y, ()) (Thack, Bhack, L At).
(14)
This operator is applied to both predicted and ground-truth fields. By encouraging alignment between Φi (n) and d(xo), the model is guided to produce predictions that are physically consistent with known partiele dynamics, while still allowing flexibility to capture unresolved biological influences. We denote this the backtrack physics operator.

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Table I: Overall evaluation results on test set.
Model
Infilling
Prediction
RMSE
MAE
RMSE
MAE
PIDM
0.538
0.493
0.491
0.446
CoCoGen
0,460
0.371
0.309
0.250
vanilla
0.410
0.330
0.306
0.245
DPS
0.393
0.311
0.298
0.238
PPIDM (ours)
0.270
0.208
0.268
0.202
4.3 Results
To evaluate the performance of PPIDM, establish four baseline comparisons of different physics integration paradigms: (i) vanilla diffusion model trained only with data fidelity loss, (ii) a physics- informed diffuuion model (PIDM) [ that incorrectly enforces the advection term the complete PDE to the training loss, (ii)) model following diffusion posterior sampling (DPS) B that injects advection information for posterior refinement only during sampling, and (iv) a model following the CoCoGen [T] framework which also uses a vanilla diffusion model and injects physics only during the last sampling steps but assumes the advection itself fully describes the system dynamics. This design of baselines isolates the effects of constraint timing of training versus sampling and physical completeness handling. We evaluate model performance using standard numerical metrics including Mean Squared Error (MSE), Root Mean Squared Error (RMSE), and Mean Absolute Error (MAE), which directly quantify deviations from the ground truth. Metries designed assess perceptual quality are not suitable for our task.
Physics Operators and Weights To demonstrate the different effects of the physics operators with varying weights, we perform a long-range infilling task across multiple spatial regions in the test set. Two representative regions are shown in Figure 2 Notably, the optimal weight and the best choice of operator varies between regions, which reflects underlying differences in local dynamics. Region B achieves more accurate and stable reconstructions under strong advection constraints, which suggests that advection strongly dominates the dynamics in this region. In contrast, region A shows better reconstruction under moderate physics guidance, likely due to more complex or biologically modulated dynamics such as phytoplankton activation or unresolved biogeochemical processes. Therefore, to achieve more accurate reconstruction of given region, region-specific calibration is needed. To ensure consistency, we use the pde operator with a fixed weight of 0.1 in all reported PPIDM results.
Comparisons of Model Performance To provide overview of model performance, we report the mean RMSE and MSE across the entire test set, excluding standard deviation due to spatially heterogencous dynamics (Table ID. To illustrate the stability of model generation, select one representative region to sample for 20 times and report the mean and standard deviation (Tables 23 We focus two tasks: long-range spatiotemporal infilling and fnture-frame prediction.
To demonstrate the consequences of treating incomplete physics fully known, the baseline which minimizes the PDE residual to zero during training results in the highest reconstruction emros.. By forcing the model to satisfy an oversimplified physics constraint during training, the method introduces conflicting gradients. The data fidelity term pulls solutions toward the true manifold Masta while the physics loss restricts them to an incorrect subspace Mphy = {x Φ(x) = 0}. This conflict corrupts the learned distribution, producing solutions that neither align with observations latent dynamics. Similarly, the CoCoGen-ban based method [T] treats partial physics as complete during inference, injecting advection constraints in the last 30% of the denoising steps. Although the early generation process is unaffected, its final reconstruction with false physics assumptions misaligns with the true dynamics. This results in globally plausible layouts but locally inconsistent details.
respect
The purely data-driven baseline achieves moderate performance. However, without physics con- straints, the model generates results that only align with the learned data distribution from the limited training samples. While the model occasionally produces plausible solutions, high variance reflects unreliable adherence to physical laws.
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Table 2: Infilling performance of baseline models and model when given only the 1st and 20th frame as input to reconstruct the intermediate 18 frames (frames 2-19). Due to space constraints, only partial frames are shown. PPIDM achieves smooth transitions that aligns best with the ground truth (GT) frames.
GT
Frame
Models
PIDM 1
CoCoGen
vanilla
DPS 3
PPIDM (ours)
2
13
19
RMSE
0.654 (0.0005)
0.461 (0. (0.173)
0.460 (0.165)
0.413 (0. 138)
0.270 (0.011)
MAE
0.608 (0.0006)
0.374 146)
0.373 (0.143)
0.333 (0.117)
0.208 (0.016)
Table 3: Prediction performance of baseline models and our model when given the first 10 frames to predict the next 10 future frames (frames 11-20). PPIDM achieves the best result in preserving the dynamics.
GT
Frame
Models
PIDM 1
CoCoGen
vanilla
DPS 3
PPIDM (ours)
17
20
RMSE
0,615 (0.0002)
0.360 (0.051)
0.348 (0.022)
0.344 (0.041)
0.266 (0.027)
MAE
0.571 (0.0002)
0.308 (0.047)
0.295 (0.015)
0.292 (0.039)
0.209 (0.022)

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DPS-based approaches B incorporate partial physics during the sampling step. Our finding confirms that integrating physics the early diffusion steps yields poor reconstruction results, because applying deterministic PDE constraints to noisy latents produces destabilizing gradient signals. As result, we follow the setup of CoCoGen [1] and restrict the integration of physics to the last 300 denoising steps. The results improve slightly compared to the vanilla diffusion model. However, we still observe suboptimal performance, as the denoising trajectory already misaligned with the physics manifold by the time guidance begins, and structural errors have accumulated that a post-hoc operator cannot correct. contrast, our PPIDM have significant improvements compared to the baseline models. achieves the lowest average MSE by guiding the reconstructions into physical solution manifolds. Although the standard deviation is moderately higher than that observed when enforcing a complete PDE, this is reasonable because using only partial physics leaves room for unknown processes and does not strictly constrain the solution.
1.2
1.2
1,0
1.0
0.8
0.8
MSR
MST.
0.6
0.4
Backtrack operator: frome
Backtrack operator: frame 10
0.2
ADR PDE opezakor: feamr:
ADR PDB opezator: froune
vanilla diffasiow: ftome?? 2
vanilla diffrsioo:: ftame 10
0.0
600
200
800
1000
sampling epoch
sompling epoch
(a) Sampling step MSE for frame near the known frame(S)
(b) Sampling step MSE for frame far from the known frame(s)
Figure 3: Reconstructing frames far from the known frames benefit more from the injected partial physics knowledge (Task: given only the 1st and 20th frame as input infill the intermediate 18 frames).
Finally, we demonstrate another key advantage PPIDM: its robustness reconstructing frames that are temporally distant from known reference frames, which is setting common in long-sequence infilling and prediction tasks. In such cases, reconstruction quality typically degrades due to the limited information propagated from the observed frames. As shown in Figure the difference in performance becomes more pronounced with temporal distance, which suggests that partial physics guidance becomes increasingly beneficial when generative uncertainty is high.
5 Conclusion and Future Work
In this paper, we present PPIDM, framework that extends chysics-informer machine learning by integrating partially known physical constraints into the training of diffusion models. Our preliminary experiments demonstrate that PPIDM outperforms both vanilla diffusion models and existing physics- informed baselines that incorrectly assume complete physical knowledge. Given the prevalence of uncertain physical knowledge across various fields, PPIDM offers a generalizable approach for incorporating such knowledge into diffusion models.
Currently, PPIDM is trained on complete time series datasets without observation noise or missing values. The method sensitive input data quality, we recommend that practitioners assess data fidelity before deployment and skip missing frames when loading data rather than performing naive temporal interpolation. Future work can focus on extending the framework to accommodate systems with uncertain or incomplete observations, such as large-scale satellite- based remote sensing data of the global ocean, which often contain substantial spatiotemporal gaps. addition, future work may explore the design of more sophisticated operators for more nuanced enforcement of physics constraints, particularly in multi-physies settings involving coupled PDEs or interacting physical subsystems.

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6 Acknowledgments
We acknowledge the research support from Duke Kunshan University. We thank the authors of the Biogeochemical Southern Ocean State Estimate dataset [21] for providing the data.
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[21] Ariane Verdy and Matthew R. Mazloff. A data assimilating model for estimating southern ocean biogeochemistry. Journal of Geophysical Research: Oceans, 122:6968 6988, 2017.
[22] Julia Wolleb, Robin Sandkühler, Florentin Bieder, Philippe Valmaggia, and Philippe C. Cattin. Diffusion models for implicit image segmentation ensembles. In Proceedings of the 5th Conference on Medical Imaging with Deep Learning (MIDL), volume 172 of Proceedings of Machine Learning Research, pages 1-13, 2022.

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Reviews:
[{"review_id": "zo6v35w847", "reviewer": "Reviewer_a71N", "summary": "The paper presents a novel framework called the Partial-Physics Informed Diffusion Model (PPIDM). The key contribution of this paper is integrating partial physical laws into the diffusion model to improve the reconstruction of chlorophyll concentration in the Southern Ocean.", "questions": "Can you explain exactly how you physically-informed transformation is implemented, is it calculating the residuals of the known parts of the PDE.", "limitations": "Yes, the authors discuss relevant limitations and future work.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 3, "strengths_and_weaknesses": "**Strengths**\n\n1. The problem examined in this article is important because there are many application scenarios, such as climate-ocean modeling, where we do not have access to accurate physical priors.\n\n2. The article summarizes the relevant work well, and the baseline is recent and competitive.\n\n**Weaknesses**\n\n1. I'm having a hard time understanding what the authors mean by *physically-informed transformation*, and what *physically-informed subspace* is, and exactly how the algorithm operates, and there is no additional information on this. If I understand correctly, the final loss is still a combination of the denoising loss and the residuals of the known physics information, which seems to be similar to PIDM.\n\n2. The authors have only experimented on the data of marine chlorophyll, and I would prefer to recommend the authors to submit the manuscript to a relevant applied professional journal where a more specialized review is available.", "was_revised": true, "final_justification": "The author's response answered my doubts and I slightly improved my score", "rebuttal": "We sincerely thank reviewer a71N for the comments and suggestions. We here provide more clarifications to the raised weaknesses and questions. \n\n***1. Clarifying Physics-Informed Transformation & Relationship to PIDM***\n\nThe physics-informed transformation applied to $x_0$ (whether ground-truth $x_0$ or the predicted $\\hat{x}_0$ at a diffusion step) is defined as $\\phi(x_0)$, where $\\phi$ represents the known physics operator. As shown in Section 4.2, $\\phi$ captures accessible dynamics, which can either be a subcomponent of a governing PDE or a partial law the system follows. For any state $x_0$, this operator projects it into a physics-informed subspace, that is, the set of states that are physically consistent under the known but incomplete dynamics, depicting the state *if only known physics governed the system*. For instance:  \n- If only the advection term is known in the ADR PDE, $\\phi(x_0)$ computes the residual of the known physics. In this case, the physics-informed subspace is defined by all states that yield zero residual under the known advection operator.\n- Fluid parcels following the continuity equation can be tracked backward along the velocity field by one timestep. $\\phi(x_0)$ computes where chlorophyll would be at the last timestep if only ocean currents (velocity field) moved it, ignoring biological growth/decay. Here, the physics-informed subspace consists of states reachable by pure velocity field alone.\n\nHowever, since $\\phi$ does *not* fully describe the system’s dynamics, minimizing the residual or enforcing strict consistency with ground-truth physics is infeasible. Instead, **PPIDM** computes $\\phi(\\hat{x}_0)$ and $\\phi(x_0)$ at each diffusion step and minimizes their adaptive, timestep-dependent discrepancy, and integrate this with the denoising loss. \n\nThis fundamentally differs from **PIDM**, which assumes *complete* physics knowledge and directly minimizes PDE residuals as a fixed loss term. PIDM’s assumption impedes learning in systems with partially unknown dynamics such as for ocean chlorophyll, where biological processes are non-negligible. **PPIDM**’s novelty lies in integrating *incomplete* physics without suppressing unmodeled dynamics, thereby enhancing generalization where PIDM struggles. Thus, while PIDM and PPIDM both combine denoising and physics losses, their mechanisms and assumptions diverge significantly. \n\n***2. Scope and Application Focus***\n\nWe thank the reviewer for the suggestion, however, we submit our paper to this venue for a wider exposure to help the broader field of researchers in the ML for science community due to:\n- Ocean chlorophyll dynamics are an ideal example for partial physics, as it is dominated by well-understood ocean currents (advection) but also significantly influenced by poorly quantified biological processes.\n- Focusing on one complex, real-world problem allows us to rigorously develop and evaluate PPIDM's core mechanics.\n- PPIDM provides a easily generalizable framework. Domain scientists can plug in their specific $\\phi$ (representing their partial knowledge) into the diffusion process using our approach.\n\nWe thank the reviewer again for the constructive feedback and hope the clarifications can help resolve issues. We will revise the final version accordingly to provide clearer explanations."}, {"review_id": "Uj3xEsNsAl", "reviewer": "Reviewer_hTSZ", "summary": "This paper introduces the Partial-Physics Informed Diffusion Model (PPIDM), a novel framework designed to integrate incomplete but known physical principles into the training of diffusion models. The primary application demonstrated is the reconstruction of ocean surface chlorophyll concentration, a system governed by both well-constrained physical processes and poorly constrained biological processes. The proposed method involves defining a \"physics operator\" that captures only the known parts of the system's dynamics. The study present experimental results on a dataset of Southern Ocean chlorophyll concentration, showing that PPIDM outperforms several baselines, including a purely data-driven diffusion model and models that enforce an incomplete physical law.", "questions": "1.The choice of the physics operator and its corresponding weight seems to be a critical hyperparameter that varies between different geographical regions. Could you elaborate on the process for selecting these hyperparameters? A more detailed sensitivity analysis across a wider range of regions would strengthen the paper.\n2.The \"backtrack physics operator\" is not clearly explained. Could you provide a more detailed mathematical derivation and a clearer explanation of how it enforces physical consistency? An illustrative example would be very helpful.\n3.Your comparison with the DPS baseline involves a significant modification to the original method. Could you provide results using a more faithful implementation of DPS? A fair and rigorous comparison to existing state-of-the-art methods is essential for demonstrating the superiority of your proposed approach.", "limitations": "1. The authors have not adequately addressed the limitations and potential negative societal impacts of their work. The \"Conclusion and Future Work\" section briefly mentions that the model is trained on complete data, but it does not explore the potential failure modes of the model when faced with noisy or incomplete real-world data.\n2. The authors should include a dedicated \"Limitations\" section and discuss the sensitivity of their model to the quality of the input data. For example, how does the model perform if the velocity field data (used in the physics operator) is noisy or has low resolution?\n3. The authors should also discuss the potential for their model to produce physically plausible but incorrect reconstructions, especially in regions where the unmodeled biological processes are dominant. This could have negative consequences if the model's outputs are used for making decisions about ecosystem management or climate policy.", "rating": 3, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 1, "originality": 2, "strengths_and_weaknesses": "Strength: \n- The core idea of using a \"physics residual difference\" rather than forcing a physics residual to zero is a sensible and well-motivated approach for problems with incomplete physical models.\n- The overall structure of the paper is logical, and the introduction provides a good motivation for the problem being addressed.\n- The application of these ideas to diffusion models for spatiotemporal reconstruction is a novel and interesting research direction.\n\nWeakness: \n- The experimental evaluation is not robust enough to fully support the paper's claims. The results are presented for only two \"representative\" regions, and the authors themselves note that the optimal choice of physics operator and its weighting varies between these regions. This suggests that the method may require significant, region-specific tuning, which undermines its generalizability. A more thorough evaluation across a much larger and more diverse set of regions is needed to demonstrate the robustness of the proposed approach.\n- The figures are not well-explained and are difficult to interpret. Figure 2, for example, presents a complex set of results with different operators and weights, but the caption and main text provide insufficient guidance for the reader to fully understand its implications. The box plots are also not clearly explained.\n- While the specific application to diffusion models is novel, the idea of incorporating partial physical knowledge into machine learning models is not new. The paper fails to adequately contextualize its contribution within the broader field of physics-informed machine learning. A more comprehensive review of related work would help to clarify the paper's unique contribution.\n- The practical significance of the work is limited by the weaknesses in the experimental evaluation. Without a more convincing demonstration of the method's robustness and generalizability, its potential impact on the field remains questionable.", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely thank reviewer hTSZ for the questions and valuable suggestions. We will add more clarifications and experiments in the final version correspondingly.\n\n***1. Visualization of representative regions and choice of hyperparameters***\n\nWe agree that regional sensitivity requires deeper analysis. Due to the difficulty to quantify physics dominance, we classify the region dynamics into two types, coastal and open ocean, where coastal regions are more under the affect of biological processes as supported by literature [0]. In addition to the initial 2-region visualization, we conducted an expanded analysis across 20 diverse regions (10 coastal, 10 open ocean). \n\nWe first show the format of our recorded data using examples of randomly sampled 2 coastal regions and 2 open ocean regions. For each region and each operator $\\phi_{\\rm back}, \\phi_{\\rm PDE}$ we trained under\n$\\mathcal L = \\mathcal L_{\\rm data} + \\lambda\\\\mathcal L_{\\rm phys} \\quad\\text{with}\\quad\\lambda \\in \\{0.001\\,0.01\\,0.1\\,1\\}$.\n\nTask: Long-range infilling of frames 2–19 given frames 0 and 20; 15 independent stochastic runs per region. \n\nIn the table, rows are named by the operator name and weight (for example, back0.001 means using the backtrack operator with weight 0.001), each cell is recorded as mean(std). We calculate the performance variance for different weights of each operator, as well as the improvement over vanilla diffusion model for the *worst* performing operator:\n| Method  |    Open Ocean Region 1 RMSE   | Open Ocean Region 1 MAE   | Open Ocean Region 2 RMSE     | Open Ocean Region 2 MAE      | Coastal Region 1 RMSE     | Coastal Region 1 MAE      |  Coastal Region 2 RMSE   |  Coastal Region 2 MAE   | \n|--|--|--|--|--|--|--|--|--|\n| back0.001| $0.303 (0.023)$ | $0.241 (0.025)$ | $0.352 (0.037)$ | $0.282 (0.023)$ | $\\mathbf{0.311} (0.025)$ | $\\mathbf{0.234} (0.031)$ | $\\mathbf{0.257} (0.029)$ | $\\mathbf{0.206} (0.026)$ | \n| back0.01 | $0.316 (0.023)$ | $0.250 (0.018)$  | $0.349 (0.039)$ | $0.280 (0.033)$ | $0.317 (0.023)$ | $0.241 (0.025)$ | $0.277 (0.025)$ | $0.224 (0.023)$ | \n| back0.1| $0.315 (0.026)$ | $0.247 (0.021)$  | $0.338 (0.027)$ | $ 0.267 (0.029)$ | $0.313 (0.016)$ | $0.235 (0.018)$ | $0.264 (0.029)$ | $0.215 (0.023)$ | \n| back1| $\\mathbf{0.297} (0.019)$ | $\\mathbf{0.235} (0.018)$ | $ \\mathbf{0.329} (0.025)$ | $ \\mathbf{0.264} (0.021)$ | $0.333 (0.026)$ | $0.251 (0.022)$ | $0.258 (0.037)$ | $0.208 (0.033)$ | \n| pde0.001| $0.320 (0.008)$ | $0.251 (0.006)$  | $0.327 (0.034)$ | $0.250 (0.018)$ | $0.331 (0.013)$ | $0.248 (0.014)$ | $\\mathbf{0.205} (0.005)$ | $\\mathbf{0.157} (0.005)$ | \n| pde0.01| $0.322 (0.012)$ | $0.258 (0.011)$  | $0.313 (0.020)$ | $0.248 (0.018)$ | $\\mathbf{0.295} (0.010)$ | $\\mathbf{0.211} (0.007)$ | $0.224 (0.013)$ | $0.171 (0.007)$ | \n| pde0.1| $0.306 (0.015)$ | $0.240 (0.012)$  | $\\mathbf{0.292} (0.020)$ | $\\mathbf{0.230} (0.015)$ | $0.328 (0.017)$ | $0.244 (0.017)$ | $0.242 (0.009)$ | $0.195 (0.008)$ | \n| pde1 | $\\mathbf{0.301} (0.018)$ | $\\mathbf{0.238} (0.016)$   | $0.317 (0.024)$ | $0.249 (0.022)$ | $0.316 (0.014)$ | $0.236 (0.018)$ | $0.242 (0.038)$ | $0.186 (0.031)$ | \n| $\\Delta_{\\text{Back}}$ (\\%) | $6.0$ | $6.0$ | $6.5$ | $6.4$ | $6.6$ | $6.8$ | $7.2$ | $8.0$ | \n| $\\Delta_{\\text{PDE}}$ (\\%)  | $6.5$ | $7.8$ | $10.7$ | $8.0$ | $10.9$ | $14.9$ | $15.3$ | $19.5$ | \n| vanilla diffusion | $0.441 (0.087)$ | $0.364 (0.084)$ | $0.463 (0.072)$ | $0.378 (0.076)$ | $0.413 (0.110)$ | $0.334 (0.109)$ | $0.402 (0.125)$ | $0.338 (0.114)$ | \n| $\\Delta_{\\text{improv}}$ (\\%)  | $27.0$ | $29.1$ | $24.0$ | $25.4$ | $19.4$ | $24.9$ | $31.1$ | $33.7$ | \n\nWe then report the best $\\lambda$ values and the mean of deltas across 20 regions:\n\n|| Coastal Region | Open Ocean Region |\n|---|---|---|\n|$\\lambda$| $(0.001,0.01)$                  | $(0.1,1)$  |\n\n| $\\overline{\\Delta_{\\text{Back}}}$ (\\%) | $\\overline{\\Delta_{\\text{PDE}}}$ (\\%) |\n|---|---|\n| $7.8$                      | $14.7$  |\n\nDue to the varying balance between physical advection and biological processes across regions, we can empirically conclude that\n\n1. Low physics-dominated areas (coastal regions) achieve their best reconstructions with small physics-loss weights, since they allow the diffusion model more freedom to capture biology.\n2. High physics-dominated areas (open ocean) perform best with larger weights, which more strongly enforce advection consistency.\n\nIn all cases, our PPIDM outperforms vanilla diffusion in both accuracy and stability by a large margin even with a small weight of the physics operator. The performance of various $\\lambda$ differs within modest ranges, so domain scientists can confidently use them as starting points and fine-tune locally if needed. \n\n***2. Figure clarity*** We will revise all captions to (a) define every marker, axis name and (b) discuss key takeaways and interpretations.\n\n***3. Contribution in physics informed ML***\nWe will extend Section 2.2 to discuss related methods that integrate partial physics, including: [1] Hybrid NN-PDE solvers correcting outputs of incomplete PDEs with NN, [2] UDEs embedding known operators while learning unknown residuals via NN, and [3] APHYNITY that uses a differentiable ODE/PDE integrator to enforce a known physics term to provide a baseline trajectory and a neural residual to explain the remaining discrepancy. However, these methods' main purpose is to *learn the PDE* under partial physics, which produce only one trajectory per initial state and thus cannot support conditioning on additional states. We will also stress our main novelty: **PPIDM is the first to (1) embed partial physics into stochastic diffusion processes, (2) support more flexible conditioning, so more robust to real world applications such as any state prediction and infilling.**\n\n***4. Mechanism of the backtrack operator***\n\nThe backtrack operator is built on the discrete continuity equation for passive tracers in a velocity field. Specifically, it computes where a given chlorophyll parcel would be at the last timestep if only ocean currents (velocity field) moved it, ignoring biological growth/decay. \n\nDerivation: Starting from the continuity equation $\\frac{\\partial c}{\\partial t} + u \\frac{\\partial c}{\\partial x} + v \\frac{\\partial c}{\\partial y} = 0,$ we use a first-order approximation of the characteristic displacement induced by advection $c(t - \\Delta t) \\approx c(t) - \\Delta t \\left( u \\frac{\\partial c}{\\partial x} + v \\frac{\\partial c}{\\partial y} \\right).$ This is equivalent to shifting chlorophyll on each point $(x, y)$ along the negative advection vector weighted by the local chlorophyll gradients: $x_{\\text{back}} = x - u \\Delta t \\cdot \\frac{\\partial c}{\\partial x}, \\quad y_{\\text{back}} = y - v \\Delta t \\cdot \\frac{\\partial c}{\\partial y}$ which enforces chlorophyll flux.\n\nExample: For chlorophyll at day 10 ($x_{10}$), $\\phi_{\\text{back}}(x_{10})$ estimates how chlorophyll would distribute at day 9 if only currents moved it (ignoring biology). Because $\\phi_{\\rm back}(x_{10})\\neq x_{9}$ (biology is missing), we do not compare $\\phi_{\\rm back}(x_{10})$ to the true $x_{9}$. Instead, at each training step we apply $\\phi_{\\rm back}$ to both the predicted $\\hat x_{10}$ and the true $x_{10}$, and minimize their difference. This penalizes any discrepancy in how the model transports chlorophyll.\n\n***4. DPS implementation***\n\nOur implementation of DPS strictly follows their official implementation with only modification in the measurement: 1. We replace the original imaging operator $A(\\cdot)$ with our proposed physics operator $\\phi(\\cdot)$. 2. We set $y = \\phi(x_0) $ as the measurement, in place of the original y. We train the model without physics and during sampling at each timestep we first apply the Gaussian denoise step, then compute $\\nabla_x \\||y - \\phi(\\hat{x}_0)\\||^2$ and take a small gradient step, which is identical to Chung et al.\n\n***5. Limitation***\n\nWe thank the reviewer for raising these important points. In a new Limitations section, we will state explicitly that our PPIDM is trained and evaluated on fully resolved, noise‑free velocity fields. We will recommend domain scientists to assess input‐data quality before deploying our model. Failure modes includes high frequency noise and large temporal gaps. Therefore, if dealing with highly incomplete physics data, we recommend (i) pre‑processing velocity fields with simple low‑pass filters before applying $\\phi$, and (ii) skipping missing frames in the data loader rather than naively interpolating them. We will add warning against over-reliance on model outputs especially in extreme biological events and recommend pairing our work with domain expert insights when used for policy making. \n\nWe thank the reviewer again for the insightful suggestions. \n\n[0] Mann et al., 2006. Dynamics of Marine Ecosystems: Biological - Physical Interactions in the Oceans.\n\n[1] Tathawadekar et al., 2021. Incomplete to Complete Multiphysics Forecasting: A Hybrid Approach for Learning Unknown Phenomena. Data‑Centric Engineering, 4(31), e27.\n\n[2] Rackauckas et al., 2020. Universal Differential Equations for Scientific Machine Learning. \n\n[3] Yin et al., 2021. Augmenting Physical Models with Deep Networks for Complex Dynamics Forecasting. ICLR 2021/J. Stat. Mech., 2021(12)."}, {"review_id": "92gSIFpP5o", "reviewer": "Reviewer_vs2C", "summary": "This paper proposes a novel Partially-Physics Informed Diffusion Model (PPIDM) for reconstructing ocean chlorophyll concentration fields when only partial physical laws are known.The model incorporates known physical processes (e.g., advection) through designed physics operators, which participate in the loss function by combining residual differences between real and predicted samples in physical projection space with data loss. Meanwhile, unknown dynamics (e.g., biological processes) are treated as noise, avoiding existing methods' over-reliance on complete physical constraints. Experiments demonstrate that PPIDM significantly outperforms baseline methods (e.g., PIDM, CoCoGen) in reconstruction accuracy and stability, particularly excelling in filling temporal gaps and predicting future values.", "questions": "Please refer to the weakness.", "limitations": "Yes.", "rating": 5, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 4, "originality": 3, "strengths_and_weaknesses": "**Strength**\n1. The integration of physical laws with uncertainties in ML algorithms is indeed understudied. The authors give a good result to solve this important problem. I really appreciate this work.\n2. The approach guides the generative model through physical residuals rather than enforcing full physical constraints, which better aligns with real-world scenarios where many definitive laws remain undetermined.\n3. The model demonstrates practical significance, particularly in its scientific value for analyzing Southern Ocean chlorophyll distributions.\n4. The experimental design rigorously considers both physics-dominated and non-physics-dominated scenarios, with results showing substantial improvements over comparable models.\n\n**Weakness**\n1. How to design the model architecture of $\\phi$? How to ensure that $\\phi$ extracts and constrain the learning process under partial-known law?\n2. The sensitivity study of loss balance factor $\\lambda$ is missing.", "was_revised": true, "final_justification": "The author's response has effectively addressed my concerns. Although other reviewers raised some doubts, in many AI for Science problems, we only know partial physical laws. The author has provided a concise and effective solution, which is why I maintain a positive attitude toward the paper. Considering that the comments from the other reviewers are also constructive, I recommend a score of weak accept or accept for this paper.", "rebuttal": "We sincerely thank reviewer vs2C for the appreciation in our work and the insightful questions. We will add the corresponding clarifications and experiments in the final version.\n\n***1. Design of $\\phi$***\n\nThe physics operator $\\phi$ captures accessible dynamics and is designed to reflect the evolution or constraint imposed by the known part of the physics. Depending on the available knowledge, domain scientists can design the $\\phi$ of their own field by plugging in the state of interest $x_0$ into a subcomponent of a governing PDE or a partial law the system follows. Specifically, \n\n**(a) Subcomponent of a governing PDE**\n\nIn many systems, the dynamics are partially described by a PDE of the form:\n$\\frac{\\partial x}{\\partial t} = \\mathcal{P}{\\text{known}}(x) + \\mathcal{P}{\\text{unknown}}(x).$\nIn this case, $\\phi(x_0)$ computes the residual of $\\mathcal{P}{\\text{known}}$, which is $\\phi(x_0) = \\left| \\frac{\\partial x_0}{\\partial t} - \\mathcal{P}{\\text{known}}(x_0) \\right| $ to measure how much the system deviates from what would be expected if only the known physics were active. \n\nIf for the predicted $\\hat{x}_0$, $\\phi(\\hat{x}_0)$ returns a large residual, the model’s output is inconsistent with known physics. However, since the system also evolves under unknown processes $\\mathcal{P}{\\text{unknown}}$, the residual $\\phi(\\hat{x}_0)$ alone cannot be minimized directly or interpreted as absolute error. Instead, we minimize the discrepancy between $\\phi(\\hat{x}_0)$ and $\\phi(x_0)$ across diffusion steps, which encourages the model to stay consistent with the known physics to the same extent as the real data, without over-constraining it where the dynamics are incomplete.\n\n**(b) Partial law of the system**\n\n$\\phi$ can also encode domain specific laws such as mass conservation laws, fluid transport, or geometric constraints. These laws may not define the full system evolution, but they represent structural properties the state should approximately satisfy. \n\n1. If the law is evolution-based like in our chlorophyll parcel transportation application, $\\phi(x_0)$ transforms a state $x_0$ into its physically consistent counterpart at another timestep, such as the previous location of a fluid parcel if only known transport governed its motion. In this case, $\\phi(\\hat{x}_0)$ approximates what the state would have been under pure known physics, and we can constrain the model by minimizing its deviation from a reference (e.g., the backtracked ground truth).\n\n2. If the law is constraint-based, $\\phi(x_0)$ measures the degree of violation of the known constraint. In many real-world or partially known systems, these constraints are not strictly obeyed, and even the ground truth may show leakage, growth, or dissipation. Therefore, there is no fixed value for $\\phi(\\hat{x}_0)$, and can only be obtained by calculating the discrepancy from $\\phi(x_0)$.\n\nRather than enforcing strict adherence to an incomplete law, this method encourages the model to match the degree of physical consistency observed in the real system. \n\n***2. Sensitivity study of $\\lambda$***\n\nWe agree that sensitivity of the model to the physics weight $\\lambda$ is crucial. Due to the difficulty to quantify physics dominance, we split our study into two broad region types, coastal and open ocean, where coastal regions are more under the affect of biological processes as supported by literature [1]. We conduct an expanded analysis across 20 diverse regions (10 coastal, 10 open ocean). \n\nTo show the format of our recorded data, we first use examples of randomly sampled 2 coastal regions and 2 open ocean regions. For each region and each operator $\\phi_{\\rm back}, \\phi_{\\rm PDE}$ we trained under\n$\\mathcal L = \\mathcal L_{\\rm data} + \\lambda\\\\mathcal L_{\\rm phys} \\quad\\text{with}\\quad\\lambda \\in \\{0.001\\,0.01\\,0.1\\,1\\}$.\n\nTask: Long-range infilling of frames 2–19 given frames 0 and 20; 15 independent stochastic runs per region. \n\nIn the table, rows are named by the operator name and weight (for example, back0.001 means using the backtrack operator with weight 0.001), each cell is recorded as mean(std). We calculate the performance variance for different weights of each operator, as well as the improvement over vanilla diffusion model for the *worst* performing operator:\n| Method  |    Open Ocean Region 1 RMSE   | Open Ocean Region 1 MAE   | Open Ocean Region 2 RMSE     | Open Ocean Region 2 MAE      | Coastal Region 1 RMSE     | Coastal Region 1 MAE      |  Coastal Region 2 RMSE   |  Coastal Region 2 MAE   | \n|--|--|--|--|--|--|--|--|--|\n| back0.001| $0.303 (0.023)$ | $0.241 (0.025)$ | $0.352 (0.037)$ | $0.282 (0.023)$ | $\\mathbf{0.311} (0.025)$ | $\\mathbf{0.234} (0.031)$ | $\\mathbf{0.257} (0.029)$ | $\\mathbf{0.206} (0.026)$ | \n| back0.01 | $0.316 (0.023)$ | $0.250 (0.018)$  | $0.349 (0.039)$ | $0.280 (0.033)$ | $0.317 (0.023)$ | $0.241 (0.025)$ | $0.277 (0.025)$ | $0.224 (0.023)$ | \n| back0.1| $0.315 (0.026)$ | $0.247 (0.021)$  | $0.338 (0.027)$ | $ 0.267 (0.029)$ | $0.313 (0.016)$ | $0.235 (0.018)$ | $0.264 (0.029)$ | $0.215 (0.023)$ | \n| back1| $\\mathbf{0.297} (0.019)$ | $\\mathbf{0.235} (0.018)$ | $ \\mathbf{0.329} (0.025)$ | $ \\mathbf{0.264} (0.021)$ | $0.333 (0.026)$ | $0.251 (0.022)$ | $0.258 (0.037)$ | $0.208 (0.033)$ | \n| pde0.001| $0.320 (0.008)$ | $0.251 (0.006)$  | $0.327 (0.034)$ | $0.250 (0.018)$ | $0.331 (0.013)$ | $0.248 (0.014)$ | $\\mathbf{0.205} (0.005)$ | $\\mathbf{0.157} (0.005)$ | \n| pde0.01| $0.322 (0.012)$ | $0.258 (0.011)$  | $0.313 (0.020)$ | $0.248 (0.018)$ | $\\mathbf{0.295} (0.010)$ | $\\mathbf{0.211} (0.007)$ | $0.224 (0.013)$ | $0.171 (0.007)$ | \n| pde0.1| $0.306 (0.015)$ | $0.240 (0.012)$  | $\\mathbf{0.292} (0.020)$ | $\\mathbf{0.230} (0.015)$ | $0.328 (0.017)$ | $0.244 (0.017)$ | $0.242 (0.009)$ | $0.195 (0.008)$ | \n| pde1 | $\\mathbf{0.301} (0.018)$ | $\\mathbf{0.238} (0.016)$   | $0.317 (0.024)$ | $0.249 (0.022)$ | $0.316 (0.014)$ | $0.236 (0.018)$ | $0.242 (0.038)$ | $0.186 (0.031)$ | \n| $\\Delta_{\\text{Back}}$ (\\%) | $6.0$ | $6.0$ | $6.5$ | $6.4$ | $6.6$ | $6.8$ | $7.2$ | $8.0$ | \n| $\\Delta_{\\text{PDE}}$ (\\%)  | $6.5$ | $7.8$ | $10.7$ | $8.0$ | $10.9$ | $14.9$ | $15.3$ | $19.5$ | \n| vanilla diffusion | $0.441 (0.087)$ | $0.364 (0.084)$ | $0.463 (0.072)$ | $0.378 (0.076)$ | $0.413 (0.110)$ | $0.334 (0.109)$ | $0.402 (0.125)$ | $0.338 (0.114)$ | \n| $\\Delta_{\\text{improv}}$ (\\%)  | $27.0$ | $29.1$ | $24.0$ | $25.4$ | $19.4$ | $24.9$ | $31.1$ | $33.7$ | \n\nWe then report the best $\\lambda$ values and the mean of deltas across 20 regions:\n\n|| Coastal Region | Open Ocean Region |\n|---|---|---|\n|$\\lambda$| $(0.001,0.01)$                  | $(0.1,1)$  |\n\n| $\\overline{\\Delta_{\\text{Back}}}$ (\\%) | $\\overline{\\Delta_{\\text{PDE}}}$ (\\%) |\n|---|---|\n| $7.8$                      | $14.7$  |\n\nDue to the varying balance between physical advection and biological processes across regions, we can empirically conclude that\n\n1. Low physics-dominated areas (coastal regions) achieve their best reconstructions with small physics-loss weights, since they allow the diffusion model more freedom to capture biology.\n2. High physics-dominated areas (open ocean) perform best with larger weights, which more strongly enforce advection consistency.\n\nIn all cases, our PPIDM outperforms vanilla diffusion in both accuracy and stability by a large margin even with a small weight of the physics operator. The performance of various $\\lambda$ differs within modest ranges, so domain scientists can confidently use them as starting points and fine-tune locally if needed.  \n\nWe thank the reviewer again for the appreciation and hope this clarifies their questions. \n\n[1] Mann et al., 2006. Dynamics of Marine Ecosystems: Biological - Physical Interactions in the Oceans."}, {"review_id": "AnZz2EgmIp", "reviewer": "Reviewer_vCAo", "summary": "The authors note that the physics only defines some of the behavior in their equations, and thus dx/dt = Physics(x) + Biology(x). From this they devise the partial physics-informed diffusion model, which regularizes that dx/dt is approximately Physics(x), but does not try to over constrain towards this assumption because it's know the physics is an incomplete picture. They test against vanilla machine learning models and models that constrain directly on the assumption that dx/dt = Physics(x) and show that this partial assumption outperforms both cases.", "questions": "1. What is the performance of this method against other partial physics methods, in particular physics-informed neural networks, universal differential equations, and physics-informed neural operators?\n2. If non-Gaussian noise is applied to the data, how does that effect the performance of the method?\n3. The method seems heavily reliant on hyperparameters of how much weight to put on the physics. Can some graphs be included to demonstrate that this is not something that requires much tuning, i.e. does this only work in a specific range of physics weights of does pretty much any value do well?", "limitations": "Yes", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "The idea makes sense, the method does well in the benchmarks, but the testing baselines are very weak. There are many other models that can include partial physics, for example physics-informed neural networks and universal differential equations are the two most popular approaches, yet neither of those are benchmarked with the baseline. \n\nAdditionally, much of the method relies on an assumption of Gaussian noise in the derivation, but biological noise is not likely to be Gaussian. For smaller particle numbers it's generally noise to follow a Poisson process, which is only Gaussian-like in the limit as the number of particles approaches infinity. However, no testing is done to see whether non-Gaussian noise in the data causes issues with the chosen method.", "was_revised": true, "final_justification": "There are no benchmarks to actually determine how good the method is, so it is not clear how it performs against the state of the art.", "rebuttal": "We sincerely thank reviewer vCAo for the insightful questions. We will include clarifications and experiments in the final version accordingly.\n\n***1. Performance against partial physics baselines***\n\nPINN, PINO, and UDE are valuable physics-informed approaches. Respectfully, however, PINN, PINO and UDE have fundamental differences to our PPIDM in objective and capabilities, so we believe direct numerical comparisons are not suitable in our case:\n\n- PINN is a PDE solver that embed the *exact* PDE residual into a single neural network to solve or infer one deterministic field. In chlorophyll dynamics, they require a fully known ADR PDE including diffusion and reaction terms (biology) which are inaccessible.\n- PINO learns the solution operator of parametrized PDEs by combining sparse solution data with exact physics residuals at collocation points. They assume complete knowledge of the governing operator and cannot enforce unknown physics terms.\n- UDE augment known dynamics with a neural residual and integrate forward from an initial condition. During inference, they still require the known physics of the state to be predicted, which is inaccessible in real world. They produce only one trajectory per initial state and thus cannot support conditioning on additional states.\n\nBy contrast, PPIDM is a novel *generative diffusion framework* that:\n- Requires only partial physics and no explicit laws or forms for unknown dynamics.\n- Directly samples an ensemble of plausible chlorophyll fields $x_0$, and can provide multiple physics plausible samples.\n- Enables more flexible conditioning on additional states, and can perform any state prediction and infilling.\n\n***2. Gaussian vs Poisson noise***\n\nThe reviewer has made an important point. In our work, we took a log on the chlorophyll data before normalization and training, as biology in the ocean follows a *log normal* distribution as supported by [1]. We will provide clarifications and make this transformation more explicit in the final version. This \"Gaussianizes\" the data and aligns well with the diffusion model’s assumption of Gaussian perturbations in the forward and reverse processes. \n\nFor experimenting with Poisson noise, to avoid disrupting diffusion model's Gaussian noising assumptions, we retain the standard gaussian schedule for learning data distribution and modify the physics loss term to a Poisson negative log likelihood summed over all spatial positions $i$:\n\n$\\mathcal{L}_{\\mathrm{Pois}}(r,k) = \\sum{i}\\Bigl[r_i - k_i\\log r_i + k_i\\log k_i - k_i\\Bigr]$, where \n\n$r = \\phi(\\hat{x}_0)$ is the model’s predicted rate, and $k = \\phi(x_0)$ is the observed count at each diffusion step. We implemented this with the same weight for each operator and observed an average of 42.3% increase in RMSE and 45.7% increase in MAE. We believe this degradation arises because the diffusion backbone remains optimized for Gaussian noise dynamics, so imposing a Poisson‐based loss partially misaligns the learned perturbation structure.\n\n***3. Choice of $\\lambda$***\n\nFollowing the rebuttal rules, we apologize for not being able to show graphs and will use tables instead. Due to the difficulty to quantify physics dominance, we split our study into two broad region types, coastal and open ocean, where coastal regions are more under the affect of biological processes as supported by literature [2]. We conduct an expanded analysis across 20 diverse regions (10 coastal, 10 open ocean). \n\nTo show the format of our recorded data, we first use examples of randomly sampled 2 coastal regions and 2 open ocean regions. For each region and each operator $\\phi_{\\rm back}, \\phi_{\\rm PDE}$ we trained under\n$\\mathcal L = \\mathcal L_{\\rm data} + \\lambda\\\\mathcal L_{\\rm phys} \\quad\\text{with}\\quad\\lambda \\in \\{0.001\\,0.01\\,0.1\\,1\\}$.\n\nTask: Long-range infilling of frames 2–19 given frames 0 and 20; 15 independent stochastic runs per region. \n\nIn the table, rows are named by the operator name and weight (for example, back0.001 means using the backtrack operator with weight 0.001), each cell is recorded as mean(std):\n| Method  |    Open Ocean Region 1 RMSE   | Open Ocean Region 1 MAE   | Open Ocean Region 2 RMSE     | Open Ocean Region 2 MAE      | Coastal Region 1 RMSE     | Coastal Region 1 MAE      |  Coastal Region 2 RMSE   |  Coastal Region 2 MAE   | \n|--|--|--|--|--|--|--|--|--|\n| back0.001| $0.303 (0.023)$ | $0.241 (0.025)$ | $0.352 (0.037)$ | $0.282 (0.023)$ | $\\mathbf{0.311} (0.025)$ | $\\mathbf{0.234} (0.031)$ | $\\mathbf{0.257} (0.029)$ | $\\mathbf{0.206} (0.026)$ | \n| back0.01 | $0.316 (0.023)$ | $0.250 (0.018)$  | $0.349 (0.039)$ | $0.280 (0.033)$ | $0.317 (0.023)$ | $0.241 (0.025)$ | $0.277 (0.025)$ | $0.224 (0.023)$ | \n| back0.1| $0.315 (0.026)$ | $0.247 (0.021)$  | $0.338 (0.027)$ | $ 0.267 (0.029)$ | $0.313 (0.016)$ | $0.235 (0.018)$ | $0.264 (0.029)$ | $0.215 (0.023)$ | \n| back1| $\\mathbf{0.297} (0.019)$ | $\\mathbf{0.235} (0.018)$ | $ \\mathbf{0.329} (0.025)$ | $ \\mathbf{0.264} (0.021)$ | $0.333 (0.026)$ | $0.251 (0.022)$ | $0.258 (0.037)$ | $0.208 (0.033)$ | \n| pde0.001| $0.320 (0.008)$ | $0.251 (0.006)$  | $0.327 (0.034)$ | $0.250 (0.018)$ | $0.331 (0.013)$ | $0.248 (0.014)$ | $\\mathbf{0.205} (0.005)$ | $\\mathbf{0.157} (0.005)$ | \n| pde0.01| $0.322 (0.012)$ | $0.258 (0.011)$  | $0.313 (0.020)$ | $0.248 (0.018)$ | $\\mathbf{0.295} (0.010)$ | $\\mathbf{0.211} (0.007)$ | $0.224 (0.013)$ | $0.171 (0.007)$ | \n| pde0.1| $0.306 (0.015)$ | $0.240 (0.012)$  | $\\mathbf{0.292} (0.020)$ | $\\mathbf{0.230} (0.015)$ | $0.328 (0.017)$ | $0.244 (0.017)$ | $0.242 (0.009)$ | $0.195 (0.008)$ | \n| pde1 | $\\mathbf{0.301} (0.018)$ | $\\mathbf{0.238} (0.016)$   | $0.317 (0.024)$ | $0.249 (0.022)$ | $0.316 (0.014)$ | $0.236 (0.018)$ | $0.242 (0.038)$ | $0.186 (0.031)$ | \n| $\\Delta_{\\text{Back}}$ (\\%) | $6.0$ | $6.0$ | $6.5$ | $6.4$ | $6.6$ | $6.8$ | $7.2$ | $8.0$ | \n| $\\Delta_{\\text{PDE}}$ (\\%)  | $6.5$ | $7.8$ | $10.7$ | $8.0$ | $10.9$ | $14.9$ | $15.3$ | $19.5$ | \n| vanilla diffusion | $0.441 (0.087)$ | $0.364 (0.084)$ | $0.463 (0.072)$ | $0.378 (0.076)$ | $0.413 (0.110)$ | $0.334 (0.109)$ | $0.402 (0.125)$ | $0.338 (0.114)$ | \n| $\\Delta_{\\text{improv}}$ (\\%)  | $27.0$ | $29.1$ | $24.0$ | $25.4$ | $19.4$ | $24.9$ | $31.1$ | $33.7$ | \n\nWe then report the best $\\lambda$ values and the mean of deltas across 20 regions:\n\n|| Coastal Region | Open Ocean Region |\n|---|---|---|\n|$\\lambda$| $(0.001,0.01)$                  | $(0.1,1)$  |\n\n| $\\overline{\\Delta_{\\text{Back}}}$ (\\%) | $\\overline{\\Delta_{\\text{PDE}}}$ (\\%) |\n|---|---|\n| $7.8$                      | $14.7$  |\n\nDue to the varying balance between physical advection and biological processes across regions, we can empirically conclude that\n\n1. Low physics-dominated areas (coastal regions) achieve their best reconstructions with small physics-loss weights, since they allow the diffusion model more freedom to capture biology.\n2. High physics-dominated areas (open ocean) perform best with larger weights, which more strongly enforce advection consistency.\n\nIn all cases, our PPIDM outperforms vanilla diffusion in both accuracy and stability by a large margin even with a small weight of the physics operator. The performance of various $\\lambda$ differs within modest ranges, so domain scientists can confidently use them as starting points and fine-tune locally if needed. \n\nWe thank the reviewer again for the important questions.\n\n[1] Koch, A. L. 1966. The Logarithm in Biology. Journal of Theoretical Biology, 12, 276–290.\n\n[2] Mann et al., 2006. Dynamics of Marine Ecosystems: Biological - Physical Interactions in the Oceans."}]

Author rebuttal:

Meta-review: This paper presents the Partial-Physics Informed Diffusion Model (PPIDM), which integrates incomplete but known physical principles into diffusion models. The application to ocean surface chlorophyll reconstruction is both timely and relevant, as many scientific domains are governed by partially understood dynamics. The rebuttal strengthened the work with additional experiments, clarifications, and a discussion of limitations. Overall, the paper makes a valuable and practical contribution to AI for Science by addressing scenarios with incomplete physical laws. Despite some limitations in benchmarking, the work is well motivated and demonstrates meaningful advances.
Decision: Accept (poster)