Generating Full-field Evolution of Physical Dynamics from Irregular Sparse Observations
Authors: Panqi Chen
Yifan Sun
Lei Cheng
Yang Yang
Weichang Li
Yang Liu
Weiqing Liu
Jiang Bian
Shikai Fang
Venue: NeurIPS 2025 poster ()
Keywords: tensor decomposition
generative models
physical field reconstruction
Primary area: machine_learning_for_sciences

Abstract: Modeling and reconstructing multidimensional physical dynamics from sparse and off-grid observations presents a fundamental challenge in scientific research. Recently, diffusion-based generative modeling shows promising potential for physical simulation. However, current approaches typically operate on on-grid data with preset spatiotemporal resolution, but struggle with the sparsely observed and continuous nature of real-world physical dynamics. To fill the gaps, we present SDIFT, Sequential DIffusion in Functional Tucker space, a novel framework that generates full-field evolution of physical dynamics from irregular sparse observations. SDIFT leverages the functional Tucker model as the latent space representer with proven universal approximation property, and represents sparse observations as latent functions and Tucker core sequences. We then construct a sequential diffusion model with temporally augmented UNet in the functional Tucker space, denoising noise drawn from a Gaussian process to generate the sequence of core tensors.
  At the posterior sampling stage, we propose a  Message-Passing Posterior Sampling mechanism, enabling conditional generation of the entire sequence guided by observations at limited time steps. We validate SDIFT on three physical systems spanning astronomical (supernova explosions, light-year scale), environmental (ocean sound speed fields, kilometer scale), and molecular (organic liquid, millimeter scale) domains, demonstrating significant improvements in both reconstruction accuracy and computational efficiency compared to state-of-the-art approaches.

Generating Full-field Evolution of Physical Dynamics from Irregular Sparse Observations
Panqi Chen'
Yifan Sun' Lei Chengl.2 2.2 Yang Yang3
Weichang Li! Weiqing Liu' Jiang Bian' Shikai Fang
Yang Liu
1 College of Information Science and Electronic Engineering, Zhejiang University
2 Zhejiang Provincial Key Laboratory of Multi-Modal Communication Networks and Intelligent Information Processing College Computer Science and Technology. Zhejiang University Miirrssoft Research Asia
Abstract
Modeling and reconstructing multidimensional physical dynamics from sparse and off-grid observations presents a fundamental challenge in scientific research. Recently, diffusion-basee generative modeling shows promising potential for physical simulation. However, current approaches typically operate on on-grid data with preset spatiotemporal resolution, but struggle with the sparsely ob- served and continuous nature of real-world physical dynamics. To fill the gaps, we present SDIFT, Sequential DIffusion in Functional Tucker space, novel framework that generates full-field evolution of physical dynamics from irreg- ular sparse observations. SDIFT leverages the functional Tucker model as the latent space representer with proven universal approximation property, and rep- resents observations latent functions and Tucker core sequences. We then construct a sequential diffusion model with temporally augmented UNet in the functional Tucker space, denoising noise drawn from Gaussian process to gen- erate the sequence core tensors. At the posterior sampling stage, we propose dessage-Passsing Posterior Sampling mechanism, enabling conditional gener- ation of the entire sequence guided by observations at limited time steps. We validate SDIFT three physical systems spanning astronomical (supernova plosions, light-year scale), environmental (ocean sound speed fields, kilometer scale), and molecular (organic liquid, millimeter scale) domains, demonstrating significant improvements in both reconstruction accuracy and computational effi- ciency compared otate-of-the-art approaches. The code available https: 7/github. com wot..////ooeeSSAAA////111111
1 Introduction
Modeling and reconstructing spatiotemporal physical dynamics in the real world has been long- standing challenge in science and engineering. In fields such aeronautics, oceanography, and climate simulation, we often need infer the evolution of physical dynamics in continuous space- time from sparse observations. For example, meteorologists must predict weather patterns from limited station data, while ocean scientists reconstruct underwater current dynamics from sparse sensors. Traditional numerical methods have been widely applied, but they typically require sub- stantial computational resources and struggle to ensure fine-scale prediction in complex scenarios.
Recently, generative models particularly diffusion models [M. D & AL S 6] - have achieved remark- able success in computer vision and related fields. These models have demonstrated strong capa-
"Correspond to lei_cheng @ zju.edu.cn, fsk@ ssk@zju.ddu.cn
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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bilities in generating high -uulliuaity images, videos, and other complex data structures. An increasing number of works have begun to explore the use of generative models for physical simulation, such as modeling underwater sound speed, turbulence, and other complex phenomena, addressing tasks like PDE solving, super-resolution, and reconstruction [+] 15, Kl].
A typical paradigm in physical diffusion modeling involves constructing an autoencoder-like map- ping from the observation space to a latent space, using visual feature extractors such as convolu- tional neural networks[[]] (CNNs) or Vision Transformers[8] (ViTs) as encoder/decoder architec- tures, diffusion model is then trained in the latent space. During inference/sampling, diffusion models use the likelihood of observations as guidance and generate target dynamics via diffusion posterior sampling[E] (DPS), which are then mapped back to the observation space. While this paradigm has shown success, it also faces several limitations in woal-world physical systems. For instance, physical observations are typically obtained from sparse sensors that sample irregularly across space and time, and are often corrupted by noise conditions under which visual-based encoders are not suitable. Moreover, the goal of posterior sampling is to generate the full-field evolution of dynamics, including inference at arbitrary continuous spatial/temporal coordinates, yet observations are only available at a few discrete time steps making standard DPS ineffective.
To bridge these critical gaps, we introduce SDIFT (Sequential Diffusion in Functional Tucker Space), a generative framework that reconstructs continuous physical dynamics from sparse, irregular obser- vations. SDIFT uses a Functional Tucker Model (FTM) to encode data into structured latent func- tions and core tensors. We prove that the FTM universally approximates complex, multidimensional physical systems. Building upon this latent representation, SDIFT applies Gaussian Processbased Sequential Diffusion (GPSD) to produce temporally coherent core sequences at arbitrary time res- olutions. Crucially, we propose novel Message- blesage-Passing Diffusion Posterior Sampling (MPDPS) mechanism that enhances conditional generation capabilities, effectively propagating observation- driven guidance across the entire spatiotemporal sequenceeven at unobserved timesteps. We evaluate SDIFT on three diverse physical domainssuperrova explosions, ocean sound speed, and active par- ticle systemsand demonstrate its superior accuracy, noise robustness, and computational efficiency compared to stateoftheart methods, even under extreme sparsity and irregular sampling, underscor- ing its broad applicability and practical relevance.
Our contributions can be succinetly summarized as follows: 1) We propose SDIFT, a novel genera- tive framework designed to reconstruct multidimensional physical dynamics from sparse, irregularly observed data, significantly advancing current methodologies. 2) We establish the theoretical foun- dation of the FTM universal latent representer and integrate it within Gaussian process-based diffusion model to achieve robust and flexible generation of latent core sequences. 3) We introduce MPDPS, an innovative posterior sampling method, facilitating effective message propagation from sparse observations, ensuring robust reconstruction across temporal domains. 4) We demonstrate state-of-the- performance across three distinct physical phenomena at various scales, highlight- ing the framework's 's generalizability and robustness. This work paves the way for more flexible, accurate and computationally efficient reconstructions of real-world physical systems, with broad implications for simulation and prediction in complex scientific and engineering tasks.
Preliminary
2.1 Tensor Decomposition
Standard tensor decomposition operates on a K-mode tensor y E R x.xxx....... where the k-th mode has I3 nodes, and each entry Y1 is indexed by K-tuple i = (i1. denotes the index of the node along the mode k. The classical Tucker decomposition utilizes group of latent factors {U* E x Jk= and a core tensor W E RB x... x- ftx, known as the Tucker core tensor, to formulate a compact representation of the tensor, where {Re K is the preset rank for latent factors in each mode. Then, each entry is modeled as the multi-linear interaction of involved latent factors and the core tensor:
i..... ii), where is
Rg
In vec(W))T(u) @ .... @ U @
re
(I)
TK

---

Training
Tunable Reconstruct flow
B x
Bx
GP-based Seq. Diffusion
Limited off-grid training data
Functional Tucker Model
continuous sequence diffusion step
Post. Sampling
Message-Pand Passing Diffusion Posterior Sampling
Sourse observations at arbilrary spatiotemporll coordinates
Learned Latent Functions
Figure Graphical illustration of the proposed SDIFT with MPDPS mechanism.
where u R is the i column of U corresponding the latent factor of node ig in mode k vec(-) is the vectorization operator, and @ is the Kronecker product. "r..rrr is the (r1.-.,, TK rk)-th element of W and ui is the rą-th element of u² (ix).
Functional tensor is the generalization of tensor to functional domain. Standard tensors work with data defined on fixed grids, where each mode has a finite number of nodes and entries are indexed discretely. In contrast, functional tensors view the entries as samples from multivariate function defined over continuous domains. This function can be decomposed into mode-wise components. Using the Tucker format, this idea leads the functional Tucker decomposition [T]] 12], which is expressed as:
(i) x vec(M)TT(u 8 (ix) 8.... & ux (ix)),
where (f1, iK) IR denotes the index tuple, and u²):: R Ra is the latent vector- valued function of mode The continuous nature of functional tensor makes it suitable for many real-world applications with nigh-dimensiooal and real-valued coordinates, like the climate and ge- ographic data.
2.2 Diffusion Models and Posterior Sampling
Diffusion models|[I D, [ are class of generative models, which involve forward diffusion process and reverse generation process. The forward diffusion process gradually adds Gaussian noise to the data, and the reverse process is learnable process that generates the data by sequentially denoising noise sample. Follow the notation in the EDM framework [E],, the probabilistic flow that describes process from normal distribution with variance a (8)) ) to target distribution p(x; (sn)) can be formulated as the following ordinary differential equation (ODE):
dx ( Vx log (s))
where Vx log log (s)) is known the score function and can be estimated by denoiser Dx(x, a (s)), ie., Vx log as)) z (Do(x; (s)) xs)= o(s) is the schedule function that controls the noise level with respect diffusion step s, and is set to o(s) s in the EDM framework.
In many real-world applications, is not directly observable and can only collect the measure- ments y with likelihood p(y xx) A(x), where A(.) is the measure operator and is the identity matrix. We aim to infer sample from the posterior distribution p(xly). which is common practice of inverse problem in various fields. The work[[]] proposed diffusion posterior sampling (DPS), widely-used method to add guidance gradient the vanilla score function and enable conditional generation:
dx
+ V (x
(4)
3

---

Then the guidance gradient with respect to the log-likelihood function can be approximated with H
Xx, logply/>x; (s)) = Xx, Da = E²
Vx, ly - A(Do(x,)) (5)
Here, Do (xx) denotes the estimation of the denoised data at denoising step s.
3 Methodology
Problem Statement: We consider a K-mode physical full field tensor with an extra time mode y whose spatiotemporal coordinate is denoted as (i, (i).-.. ,kk L). Without loss of generality, suppose we have M offgrid observation timesteps T = 1,,,..,,, At each (ms the observation No contains Nm samples, where yi.... .fm is the entry value of y set O.... = {(imm fm Vin. A.n ) n..,
indexed at (in.. fm). We denote the entire collection of observations by O {O HM Jm=
We assume the patterns of observations at each time step varying. The goal is infer the value of underlying function yii, t) RK x R+ - R at arbitrary continuous spatiotemporal coordinates. If we treat the problem as function approximation task, it can be solved by numerical methods [13. [4] and functional tensor methods [III, [12].
However, in high-dimensional settings, directly approximating such a function from sparse observa- tions is challenging due to the curse of dimensionality ([S]]. To address this, we adopt a two-stage framework commonly used in the generative physics simulation [9, 1SL II, [Z]. In the first stage, we train a diffusion model using observations sampled from B batches of homogeneous dynamics, denoted as D = {O"] hh11 where Ob is an instance of O as mentioned before. the second stage, given observations of the target physical dynamics, we employ the pretrained diffusion model as data-driven prior and perform posterior sampling generate the full-field evolution.
Compared with existing works [LL 15), I8, KL M], our setting is unique in two key aspects: 1) the training data is both sparse and off-grid across the spatiotemporal domain; 2) the objective is to generate the full-field evolution over a continuous spatiotemporal domain rather than on a discrete mesh grid. These characteristies introduce two main challenges. First, during model training, more general and flexible latent space mapping is required to effectively represent the sparse and irregular observations. Second, during posterior inference, it is crucial leverage the limited-time- step observations as guidance to generate the continuous full-field evolution.
Therefore, propose SDIFT, novel sequential diffusion framework that integrates the functional Tucker Model, Gaussian process-based sequential diffusion, and message- message-passing diffusion poste- rior sampling. A schematic illustration of our approach is presented in Fig. III
3.1 Functional Tucker Model (FTM) as Universal Latent Representer
Without loss of generality, we consider a single batch of training data O+ = (O), , J. For notation simplicity, we omit the superscript b. Our goal is to map this sequence into structured la- tent space. To achieve this, we adapt the Functional Tucker Model (FTM), a general framework that naturally captures the inherent multi-dimensional structure of physical fields and provides compact representations well-suited for sparse or irregular scenarios.
We treat O as irregular sparse observations of K-mode functional tensor with an extra time mode V. Then, aligned with functional Tucker model (2), we assume the tensor sequence shares Ra
latent functions {foz (ix) R + I
[k=1 parameterized by K neural networks, but with dif- ferent learnable Tucker core (We.. km E IR R4 x... Rx M m==-- Specifically, for each observed entry
(i, fm Ml..... ) O,,,, it can be represented as:
Vi.f.. R vec(WW...) (c)) (11) @
fxx (ik)).
(6)
After solving the above representation tasks for all entries in O, we treat the estimated core sequence W : (W fm. Jm=1 as the latent representation of O. This process serves an encoder, analogous to that a variational autoencoder (VAE) [L9)) which maps the sparse observations to random vari- ables in a latent functional Tucker space. On the other hand, given sampled core tensors W Wi.. and the latent functions, Equation (K) functions as a decoder, allowing inference of y(i, tm)) at arbitrary spatial coordinate i.
For the completeness of the paper, we further claim and prove that the Universal Approximation Property (UAP) of FTM. Theorem 1. (UAP of FTM) Let X1, .x be compact subsets of K

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....X.k) Choose a E (X, x x XK). Then, for arbitrary é > O, there exists sufficiently large {Rı > R Rx
0, Rx > 0}, coefficients {arion rn Iri- Jk
K such that and neural networks {U Irx 3k
R
RR
S II k=1
<6.
xX(XXx-XXXX
The detailed proof of Theorem 1 is given in Appx. AL Theorem states that the functional Tucker model can approximate any function L2 over the K-dimensional input space, thereby ensuring its expressive power for representing multidimensional physical fields. Moreover, consistent with the low-rank structure in physical fields(20, 21]. the learned cores capture interactions among the K latent functions, promoting computational efficiency [[]]] and offering more interpretable rep-
resentation [22]] compared to nonlinear encoders like VAËs [19] or FiLM [23].
To train the latent functions (fe,, ) and obtain batches of core sequences (10) } B we propose
b=
to minimize following objective function over the entire training dataset 'D {00)
B
1=1
CFTM Eitttiii )~D
p1...... ve
(f)) (i1
fk (ix (ix)) STV(W]] (8)
where TV(WPP denotes total variation regularization [24|] over the core sequence on the temporal mode to ensure their temporal coherence, controlled by coefficient B. The training process is con- ducted using alternating-directii strategy that iteratively updates latent functions and the core tensor sequence.
After training, we obtain a set of K learned latent functions {fge (-)) R= that summarize universal co-
ordinate representations, and batches of learned sequence (W*) 1b=1 which capture the unique characteristics of the corresponding physical fields.
3.2 Gaussian Process based Sequential Diffusion for Core Sequence Generation
Given [W from FTM training, we aim build diffusion model that generates core se- quences with cores extracted at arbitrary timestep. For notation simplicity, we omit the super- script Assume that W = (Wt.... Wish } is drawn from an underlying continuous function W(t) R, + Rd x fxx at T {ti.--. .far J. where each element can be any continuous timestep. To model the distribution of W(t). we follow the idea of functional diffusion model [25, 26), and choose Gaussian Process (GP) noise [ILS] as the diffusion noise source. This approach naturally handles treeularly-samppe ie., assign varying noise level at different positions in the sequence, enabling the generation of temporal-continuce sequence samples.
timesteps and incorporates cemporal-correlate perturbations,
Specifically, we assume the noise sequence & {E.,.". GP(O, n(t;; 42)), where 4g) = exp (-7(1, 12) is set as the radial basis function (RBF) kernel to model the temporal correlation of noise in sequence. As vanilla GP can only model the scaler-output function, here we actually build multiple independent GPs for each element of the core.
Etsr is sampled from GP
In the forward process of diffusion, sample of core sequence W is perturbed by the GP noise During the reverse process, the noise sequence from GP is then gradually denoised by a precondi- tioned neural network D, to obtain the clean core sequence. Notably, when M 1, the model degenerates to the standard diffusion model (L). We refer to this model as Gaussian Process based Sequential Diffusion (GPSD).
Then, the training objective for the GPSD model is:
LGPSD = =E Ee) ~.E. EENTT ((a(s) (WW 6m C.WWW...C...... +(),. W.,, (9)
where X()(s)) is weighting function dependent on the noise level (s).. Note that we aim to learn the distribution of single core rather than the entire sequence of cores, allowing for flexible modeling of core sequence length and improved computational efficiency. To better capture temporal correlations within the sequence, we propose temporally augmented U-Net (details in Appx. B and summarize its training procedure Appx. D_I
TTV(W) = DM W.n.. W:...ll
5

---

4
likelhhodd
ikelihood
gradient guidance
gradient
guidance
W
W
(a) DPS: Guidance missing at t'.
(b) MPDPS Collected message as guidance at t'.
Figure 2: Illustration of DPS and MPDPS to handle the case of no observations at l'.
3.3
Message-Passing Diffusion Posterior Sampling
With pre-trained GPSD and FTM, we generate the core tensor sequence from GP noise and recon- struct dynamics at arbitrary (i, t) via FTM. We further introduce a posterior sampling method for conditional sequence generation from limited observations.
Consider the task of inferring dynamics at target time set Tear {11.... ,1M). With pre- trained FTM, this reduces to generating the core tensor sequence W (Wi,.... Wewd } from the observation set O. We assume that observations with Tobsl = L S M. Under standard DPS, only the observed cores Wabs == (W,, t E Tobs } guided by their corresponding observations Ot. while the remaining cores are generated without guidance, as illustrated in Fig. D(.). This is suboptimal, as real-world physical dynamics are temporally continuous and correlated, meaning that even limited observations can provide valuable guidance for the entire sequence.
only available subset Tobs S Trars
To enable posterior sampling of the entire core sequence, we propose Pessage-Passing Diffusion P Posterior Sampling (MPDPS). The key idea is to exploit the temporal continuity of the core sequence smooth and propagate guidance from limited observations across the full sequence, as illustrated in Fig. b).
Without loss of generality, we focus on the timestep with observation & E Tobs as a representative case, and demonstrate the propagation mechanism of how guidance message of O, pass to the residual core sequence Wt,, = W We, where [W,, = 1. At diffusion step s, the guidance message group of On, passed to Wi,, can be expressed as:
( Im Tinr = Vww, logp(Oul|W))) Vwg, log p(o 1, wpw. Jaw
(10)
where Piett. Vw denotes perturbed WAs km logp(O,|MM),) the likelihood gradient of Or respect to W. W..
at diffusion step s, Vw is the operator to compute the gradient respect to each core in W and Wi, denotes the clean core timestep 4, However, directly computing (III) is intractable. While (WO.) ) is Gaussian factor due to the multilinear structure of the FTM, the conditional distribution p(W) f: (W.)) is unknown, making the integration term intractable.
To handle this, we further assign element-wise GP prior over the core sequence, consistent with generation by GP noise in Sec. (.21, and claim that p( W (W can be estimated Gaussian factor via the denoiser network Dp and Gaussian process regression [|_]]] (GPR). Then we can work out an tractable approximation of (II) as follows:
Vwz, p(O, W.W) x Vw 1 [yr, -B,,TOW, E,, (y+, B,,T(MW,,,)) (11)
where yt, is vector collecting entries in O,,, and T(W;,) with shape of (M 1) ( R is the concatenation of vectorized estimated clean W E Wi, from Do. which takes every W ti.. W 14 as input. B,, and Σ. are constructed by the latent functions in FTM and the GPR kernel matrix in a closed form. A full derivation is provided in Appx. C Note that (ll) has quadratic form, enabling efficient gradient computation.

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At diffusion step we go through all observations, compute the guidance messages to be passed with (LT) and for core Wi, at tn Tar- the guidance messages from observations are:
ww, log Dgg(OO x0 Tow (fn) Vw, log log, + Σ Pi
(12)
in where To (tx) tf Tobe, otherwise O, and Vw: p(O, Wi. denotes the guidance
for core at timestep with direct observations, which is equal to the DPS guidance in ( Then we can plug (I22) into (2)) to perform posterior sampling.
As shown in (l12), MPDPS propagates observation-derivel guidance across the entire core sequence. For cores timesteps with direct observations, this guidance further smoothed by messages from other observed timesteps. This smoothing mechanism enhances the robustness of the generated sequence, especially under noisy or extremely sparse observations. Consequently, MPDPS can out- perform standard DPS even when Tia = Tobs. as demonstrated in our experiments. We provide a more detailed explanation of the MPDPS mechanism Appx. D to facilitate understanding. The full MPDPS procedure is summarized in Algo. M.
Algorithm 1 Message-passing Diffusion Posterior Sampling
Require: Deterministic Sampler Ds(:: (s)). noise power schedule (or(s)) observation set O, target time set Tear, observation time set Taba, kernel parameter Weights Ç.
IM } at Tear with kernel function Ki,- = 1: Generate initial GP noise sequence (W 61 W
exp (-> 1) 2: for i - ,, do
3: Estimate all denoised cores at step s;: W +- Do (W ( Vj.
4: Evaluate dV (s) at diffusion step s;: Dj + (Wi, Ww ())(/((s))
5: Euler step: W f + Wi, + (o(s)-1) o(s:)) Dj, Vj.
6: if (44-1) # 0 then
7: Secondorder conrection: W!, DD 1 ( Vj.
8: Evaluate wW/(o(()) at diffusion step 5;-1: : Dj- t (W -W., /o
9:
Apply the trapezoidal rule diffusion step 8:-1:
W i-
Wi, (a(l++1)
(I.) + {D}))
10:
end if
11:
Compute the guidance gradients defined in (122) for all target cores, and accumulate each into its corresponding core.
end for
13:
return Guided generated core sequence (Wa,,o, Wex.ol
4 Related Work
Tensor-based methods are widely used reconstruct multidimensional data by learning low-rank latent factors from sparse observations. Recent works in temporal tensor learning [27 28, 22, 29] relax the discretization constraints of classical tensors, enabling modeling of continuous temporal dynamics. Functional tensor approaches [022 III, 30, [1] go further by modeling continuity across all tensor modes via function approximation with structured latent factors. While effective, these methods are limited single-instance training and inference, and lack the ficxibility to handle sets of tensors with varying dynamics, e.g., PDE trajectories under different initial conditions.
Toward generative physical simulation, several works [9 17 I.G. E. 6 have applied traitre diffusion models, achieving notable progress across various tasks. However, most existing methods assume well-structured data and struggle with real-world reconstruction scenarios involving sparse and irregular observations. To address this, [5]] proposed modeling off-grid data using conditional neural fields (CNFs) followed by diffusion. Yet, this approach do not explicitly capture temporal continuity, limiting their ability to generate full-field dynamics. Additionally, it suffers from high computational cost due to repeated gradient backpropagation operated on CNF during posterior
7

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Supernova Explosion 1%
Ocean Sound Speed
Active Matter 1%
Methods
3%
1%
3%
3%
Observation setting 1 Tobo
Tear
Tensor-busse LRTER DEMOTE
0.558
0.044
0.429 00043
0..45 + 0,036 0.217 +: 0.066
0.302
+ 0.104
0.258 +: 0.022
[S]
1.213 0,217
0.358
0.127
0.086
0.950
0.497
NONFAT
1.229 0.127
1.197 = 0.204
0.402 - 0.090
0.330 = 0.101
0.911 = 0.457
0.867 + 0.413
Attention-buee
Senseiver [[N]]]
0.349 1: 0.023 [0.264 1: 0.037 0.2005
0.031 0.345
0.094
0.264 1: 0.076
Diffusion-based
CONFiLD [S]
0.427
0.037
0.201
0.034
0,529
0,5075 0.830
SDIFT w/ DPS
0,339
0.116
0.291
0.033
0.194
0.073
0.160
0.035
0.298
0.065
0.174 + 0.043
SDIFT w/ MPDPS
0.283 + 0.026
0.272 + 0.025
0.146 + 0.046
0.108 + 0.043
0.215 = 0.068
0.156 + 0.046
Observation setting
|Tobsl
|
Tensor-bused
LRTHR (M
0.783
+ 0.416
0.813
+ 0.296
0.610
+ 0.323
0.508
+ 0.297
0.620
+ 0.484
0.598
+ 0.527
DEMOTE
[S
1.351 1 0.209
1.223
+ 0.397
0.549
+ 0.181
0.513
0.199
1,261
4 0.614
1.277
1 0.603
NONEAT
1.278
0.214
1,254
0,2785
0.465
1 0.153
0.420
0.189
1.126
0.514
1,270
1 0.485
Attention-buse
Senseiver [0]]
Not capable
Diffasion baved
CoNFiLD (E)
0.757 +: 0.199
0.6575 +: 0.148
0,310
+ 0,054
0,2615
= 0.038
10.8265
+ 0.167
0.779
+ 0.161
SDIFT w/ DPS
0.659 0.057
0.6450
0.054
0:412
+ 0,156
0.407
0.16
0.674
0.353
0.637 + 0.1133
SDIFT MPDPS
0.163 0.335 1: 0.122
0.181
0.084
0.165
0.296
0.096
0.256 1 0.087
Table VRMSEs of the reconstruction results for all methods across three datasets, evaluated under two observation settings with different observation ratios.
sampling. []]] introduced an attention-base model for reconstructing fields from sparse, off- off-grid measurements, but it lacks explicit temporal modeling and struggles with rapidly evolving fields.
In parallel, operator learning methods G22, G have emerged as powerful tools for physical modeling by learning mappings between functions, and have been applied to field reconstruction tasks [44 35 96, BZ]. However, these methods often rely on task-specific designs and prede-
fined observation patterns, limiting their adaptability. Notably, [2 reported that FNOs can perform
poorly in high-dimensional settings with sparse data.
To address these gaps, propose SIDFT, method well-suited for high-dimensional, continuous spatiotemporal field reconstruction in sparse and irregular settings, with strong flexibility and per- formance across domains and scales.
5 Experiment
Datasets: We examined SDIFT on three real-world benchmark datasets, which span across astro- nomical, environmental and molecular scales. (1) Supernova Explosion, temperature evolution of a supernova blast wave in a compressed dense cool monatomic ideal-gas cloud. We extracted 396 records in total, each containing 16 frames, where each frame has shape of 64 x 64 64. We use 370 for training, randomly masking out 85% of the points in each record to simulate irregular sparse data. The remaining 26 records are reserved for the well/datasets/ stnttassess/upervvaaa supernova explosion 64/; (2) Ocean Sound Speed, sound speed field measurements in the pacific ocean. We extracted 1000 records of shape 24 x 38 76, us- ing 950 for training (with 90% of points randomly masked) and reserving 50 for testing. (https: //ncss.hycom www..ooyooooootthredeeocooo wwtp//oreed///css///geeeeeeeeeee http///s//////lyyyy//// (CGLLBBO.00//xppp. 08/expt 93. 25..0tts3z/dataset. httml. (3) Active Matier, the dynamics of rod-like active particles in a stokes fluid simulated via a contin- uum theory. We extracted 928 records, each of size 24 x 256 x 256, using 900 for training (with 90% of points randomly masked) and reserving 28 for testing. (tttps://polymatthttt the well/datasets//ccoo active matter/)
org/
org/
Baselines and Settings: We compared SDIFT with state-of-the-an physical field reconstruction methods and densor-based methods: (1) Senseiver [[8], an iitention-basee framework that encodes sparse sensor measurements into unified latent space for efficient multidimensional field recon- struction; (2) CoNFiLD [5)], a generative model that combines conditional neural field encoding with latent diffusion to generate high-fidelity fields. (3) LRTFR [[Л]], low-rank functional Tucker model employing factorized neural representations for tensor decomposition; (4) DEMOTE [G8],
neural diffusionreaction process model that learns dynamic factors within tensor decomposition
8

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Ground Truth Observation
(MPDPS)
(DPS)
Sencelver VAMSEI0:375
CONFILD
LATER
frame 80
frame frome frame
Figure 3: Reconstruction of Active Matter dynamics under observation setting 1 with = 1%. SDIFT guided by MPDPS produces notably robust reconstmuctions with fine-grained details.
framework; (5) NONFAT [27]], a bi-level latent Gaussian process model that estimates time-varying factors using Fourier bases; We evaluate our method against baselines on reconstructing temporal physical fields under two observation settings with observation ratio E {1%, 3%}. }. In observa- tion setting 1, Tohs Tear,
each observations occur at ratio p. In observation setting
Tobs C T for 4 E Tobs we retain observations at ratio whereas for 4, E Tear \ Tobs no observa- tions are available. We assume that Toba constructed by selecting every other element from Tear so that (Tobal
/ We replace MPDPS with traditional DPS [[]] to constitute an ablation study to showcase the effectiveness our proposed MPDPS. The performance metries the Variance-scaled Root Mean Squared Error (VRMSE, see Appx. E. for definition), which offers acale-independent way to evaluate model performance. Each experiment was conducted 10 times and we reported the average test errors with their standard deviations. We provided more implementation details in Appx. IB.
The quantitative results Tab. II demonstrate that SDIFT consistently outperforms all baselines by substantial margin under both observation settings and 2. We further observe that tensor-based methods generally lag behind training-basee approaches, since they cannot exploit batches of historical data and must reconstruct the field from extremely sparse observations. In particular, temporal tensor techniques that ignore continuously indexed modesuch as NON- FAT and DEMOTEperform poorly across all three datasets compared to LRTFR. The qualitative comparisons in Fig. indicate that SDIFT with MPDPS produces more faithful reconstructions.
Main Evaluation Results:
Observation setting 2 presents a more challenging scenario, in which all baselines exhibit signifi- cant performance degradation. CoNFiLD and SIDFT with DPS can still reconstruct via uncondi- tional generation without any observations, whereas Senseiver cannot. As shown in Fig. g. SDIFT guided by DPS generates physical fields randomly at timesteps without observations, owing to the absence of guidance in such cases. Similarly, Senseiver fails when observations are unavailable. In contrast, MPDPS effectively overcomes these limitations and consistently produces accurate recon- structions, highlighting the robustness of the proposed method. Additional unconditional genera- tion results, conditional reconstruction results and detailed analysis on CoNFiLD are provided in Appx. GHIGWER respectively. We also demonstrated the effectiveness of using GP noise as the diffusion source in Appx. G.5 These results collectively demonstrate the effectiveness of SDIFT with the MPDPS mechanism.
Sampling Speed:
We compared the sampling speed of our method with CoNFiLD [S]] on NVIDIA RTX 4090 GPU with 24 GB memory; the results are shown in Tab. D. One can see that our proposed method is significantly faster than CoNFiLD across settings. This rapid inference speed stems from two key factors. First, we formulated the reverse diffusion process solving a deterministic probability-flow ODE rather than stochastic differential equation (SDE, as in CoNFiLD), which largely reduces the number of neural network evaluations during sampling. Second, by employ-

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Ground Truth Observation
(MPDPS)
(DPS)
Senseiver
CONFILD
LATER
frame 80
frame frame frame frame
Figure 4: Reconstruction of Supernova Explosion dynamics under observation setting with P 3%. Since DPS does not provide guidance at timesteps without observations (i.e., frames # 3,9,15), SDIFT generates the corresponding physical fields randomly. In contrast, MPDPS effec- tively addresses this limitation and yields smooth reconstructions.
Supernova Explosion
Ocean Sound Speed
Active Matter
Methods
1%
3%
#Para. |0 1%
3%
#Para. p 1%
3%
#ara
CoNFLL
23M
15.9s
10M
10M
SDIFT w/ MPDPS
2.236
5.43
26M
0.84s
0.89
15M
1.31s
1.42s
12M
Table 2: Average sampling speed for reconstruction with different observation ratios observation setting 1.
ing functional Tucker model, we endowed the likelihood with a quadratic structure that enables efficient computation of posterior gradients. In contrast, CoNFiLD defines its likelihood through conditional neural field and must compute gradients via automatic differentiation at each diffusion step, significantly increasing the computation time.
Robustness against Noise: We evaluated SDIFT with DPS and MPDPS against three noise types: Gaussian, Poisson and Laplacian, with different variance levels, and results are at Tab. in Appx. Gdl which demonstrates the robustness introduced by proposed MSDPS module.
Conclusion and Future Work
We presented SDIFT, generative model that efficiently captures the distribution of multidimen- sional physical dynamics from irregular and limited training data. We also proposed novel pressage-passing diffusion posterior sampling mechanism for conditional generation with obser- vations, achieving state-of-the-ar reconstruction performance with significant computational effi- ciency. A current limitation of our work is the lack of explicit incorporation of physical laws into the modeling process. In future work, we plan integrate SDIFT with domain-specte physical knowledge, enabling more accurate long-range and wide-area physical field reconstructions.
7 Acknowledgement
This work was supported in part by the National Natural Science Foundation of China under Grant 62371418 and Grant 62322606, in part by the Fundamental Research Funds for the Central Univer- sities 226-2025-00169) and part by Zhejiang University Education Foundation Qizhen Scholar Foundation.
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Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology. scientific rigorousness, or originality the research, declaration is not required.
Answer: [NA]
Justification: The core method development in this research does not involve LLMs any important, original, or non-standard components.
Guidelines:
The answer NA means that the method development in this research does involve LLMs as any important, original, or non-standard components.
Please refer to our LLM policy ottpas///eaarrrsscccc//oorrreere for what should or should be described.
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APPEXNDIX
A Proof of Theorem 1
Here we show the preliminary lemmas and proofs for Theorem We consider the general tensor product between K Hilbert spaces.
Definition 1. foa, } is an orthonormal basis for Hilbert space H4 Vi.
Lemma 1. {Va, @ vag) } an orthonormal basis for H1 @ H2.
Proof. See Lemma 2. the Appendix A of [[9]]
Lemma 2. {Vo1 8 ... & Goge J is an orthonormal basis for Hi @ ... @ HK.
Proof. By repeatedly applying Lemma 2., we can show that (Vay 69 6 Fag } an orthonormal basis for Hi @ ... & H..
Theorem 1. (Universal Approximation Property) Let X1, Xx be compact subsets of R
Choose u E (X, x Xk). Then, for arbitrary > we can find sufficiently large Bir RR and neural networks {fr,.... " } {R > 0,-.. Rx > 0}, coefficients {an..- Jk ITL. yK
such that
R
RK
<.
(13)
Jk
TX
k=1
x2(X,x--XXx)
Proof. Let (o.,).... (rrx } be an orthonormal basis for (X)), (Xxx), respectively, Ac- cording to Lemma 2, fØr, ... Ory } forms an orthonormal basis for (2 X x... Xx). Therefore, we can find sufficiently large set {R1 0.... Rx > 0} and arbitrary e such that
Rx
Cr
II I
(14)
L2 X-XXIx
Here, Cry ox is defined as
Cri ox = =f ulii,....
K
K
(ik)
k
(ix)
II
dis.
(15)
k=1
Also, with the universal approximation theorem 40], we can find neural networks (sk JK satisfy
k=1
11 - - (1+ R&) K
Vk.
(16)
| (XX
21

---

First, we have
R,
Rx
K
1
Cri
11
TK
k=1
[2(X1x
x==xxxxx
R1
Rg
R1
R
K
Crl
Pru
k
+
Crieh PK
s
k=1
k=1
k=
xx(Xxx--xXx)
RR
Cri.
Pre
(17)
TK
LP
xIXXxx..XXXX
A1
R1
RK
K
R1
RK
K
Cr
Σ
Cra
fni
JK
P1
FK
T2
FK
LA X-X
The first term A1 can be determined by Eq.
We are estimate the second term.
R
Rx
K
R
Ri
K
Σ
J
JK
TR
k=1
FK
k=1
R
Rg
R1
Ore (oh
fth) + Σ
Cr Pk (
II
d or (o?
+
Cri
KK
2=2
k=3
B1
B2
(18)
R1
Rx
K 11
Jr MO - -SK)
Cr
JK
P1
FK
1=1
BR
Then, we have
A BB + + Bxll
(19)
< |B + + |Bx (xX
Specifically, we have
RK
K
[B (X
(bh
-fth
II
dig
FK
kmm
km
(20)
Bi
R2
Rx
K II
Cr
Fx
Pre
(
dig-
P2
FK
km
km
C2
C1
By applying Cauchy-Schare incquality, we have
R
R
K II
|
/
Σ
IC1
Σ
C22
dik
k=1
( Σ C 2 II diu
Ri
(21)
Σ
Caf
C2l
k=2
D2
Da
22

---

Thereafter, we have
D1
R Σ C1
II
K
dir
=
Σ
R1
fIGP
CP II
K
dis
Σ
R1
Σ
R2
Rg
Cri
Px
K
Oree
11
Jem
21
72
km
(2(x(33)
Since ( 1K Prrjkes k=1
are all orthonormal basis, we have
R2
Rx
K F
R2
Ra
K
Σ
hk Ore
Sr
P2
TK
2=2
12(Xxxx-xxx)
Xx)
km 3
x2(xxx..xxk)
R2
Ra
k
1 12(22)
Cri Tx
k=
(x(xxx--Xx)
R2
Ra
Rg
(23)
Cra
ore)
72
TK
Xxx
R2
$
12
TK
Substituting Eq.(23)
(2.3)
into Eq. (22)) and we can get
R Σ
K
D1
/
C1
di Σ
II
di
2=2
k=
R1
Ra
RK
R1
R2
RK
K
(24)
Cr.. cK 1 <
R
Cra. ex
2 < <[I
Re) ||
71
12
TK
2=
F1
12
FK
k=2
Next, we have
R1
/ ΣΣ
D2
C2 Pdi = Σ
10),
12.1 ddi
Rg
Σ
10h (allEA()))
(25)
Ti
2
<
12
71=
(1+ 2 RA) 7 K || (11|
k= Rs
K2 K
1/2
22
(X
Therefore, have
K ①
2
2²
||B D,D2
Rx)
(4 (4 K2 K2 ||u
4K
k=
(26)
In the similar sense, we can also prove
é
Bx 2
<
(27)
4K2
Therefore,
2
A K
(28)
4K2 2
23

---

Substituting Eq (14) and Eq Eq.28)
(28)
into Eq.
we can derive
II
A2
An
(29)
k=1
LX(Xxx--XXx)
Since dr Tk random coefficients that can take arbitrary values to approximate Cry,.. Pky have
R
Rí
K
<6
(30)
TX
k=
12(XIx--XXX)
Thus, we complete the proof.
B Illustration of the architecture of the proposed temporally augmented U-Net
w
Temporally Augmented U-Net
lmpa disc
QUERAE
(oT.C.
comv2d
reshope t0 XM1 SMM C
reshage (M (.)
convld
Figure 5: Overview of the Temporally Augmented U-Net. We make slight modification by adding a CDnvID layer to capture temporal correlations among latent features (A special case K = 3).
We propose temporally augmented U-Net by incorporating lightweight temporal enhancement module. As illustrated in Fig. we add Conv layers to capture temporal correlations among latent features without altering their original dimensionality. The proposed module requires only minor modifications and is straightforward to implement. Unlike video diffusion models that incur high computational costs by modeling full sequences [[]] our approach circumvents this issue by cnabling the modeling of frame sequences of arbitrary length and at any desired temporal resolution.
C Derivation of guidance gradient of log-likelihood
Here, we derive the guidance gradient log-liiel with respect W
Vw W log P(O, Wi.) = Vw k4 log [ [(OO., p pw wy
aw to
(31)
tern
where Gi = Vw log (Og, (i) is the likelihood gradient of O₁ respect to W W denotes perturbed W at diffusion step s, Vw,,, the operator to compute the gradient respect to each core in W (.. and D is the estimated clean target core. Note that in previous step, the latent functions of functional Tucker model are learned. Together with the observation index set specified at f;, the proposed functional Tucker model reduces linear model. Therefore, term
which corresponding to the likelihood, can be directly expressed as:
( N A e
(32)

---

where Yo, E R., is vector containing Nr, entries in O.. We collect the indexes of Nr. entries in Or, and form the index matrix 4. I x Then, A. R, xII Rs the factor matrix obtained by evaluating the index matrix I, on the Kronecker product of the learned latent functions, f0 Jox- wh E IRl
R& is the vectorized form of W
To make the following derivation clearer, we vectorize each element in W and concatenate them into a matrix, denoted as W E R M-I Ra, As for term 2, we have
p(W. D. p(w (W!) = / p(w W (WW In
W in ddw
No
(33)
Ewo, wp(W / (p(w? w (1,))
We approximate the term 2 with
Ew WW, wi, (lp( W (.,) p(w? / E| 14 (..)
(34)
which closely related to the Jensen's inequality. The approximation error can be quantified with the Jensen gap.
Note that EDM[], the denoiser Da(-) can direct approximate the clean cores:
w ((W)))
(35)
where T(W ) with shape of M - 1 W
k=1 R& is the concatenation of vectorized estimated clean fm E Ww, from Dg, which takes every Wi,,. Wi,, input. Specifically, assume W is
IT
the i-th element of Wiy then:
((W))) = vo (s)))
i = vec(
(36)
The term now can be estimated using
p(w?, W = p(w, T(W
(37)
With similar rationale in the training process the GPSD, we impose an element-wise Gaussian process prior over the estimated clean cores Wihh Under this prior, the conditional distribution in (67) given by:
p(w? W N Er,),
M1 = TT
K-
Tell Tit
(38)
E,, = Kee.
KT
k
I.
where T/ denotes the residual target time set, obtained by excluding t from Tear-
Until now, we have
Vw: i4
(og
= plya W ddw?
(39)
f ( Iw Jdw? = E.
By combining (32) and (8),
F. also has closed-form distribution:
E ~ Nye,, A " E,, A (y (.W)..
(40)
where B,, Ar
K
and Er, = 22I + A Er, AT, for abbreviation.
The
Tit T2
25

---

D Algorithm
D.1 Training phase
In our setting, the training data irregular and sparse observations sampled from B batches of ho- mogeneous physical dynamics at arbitrary timesteps. Given batches of training data, we first use the FTM model to map these irregular points into shared set of K latent functions and corresponding core sequence batches. The K latent functions are then fixed, and GPSD is used to learn the distri- bution of the core sequences. Our model flexible and well-suited for handling irregular physical field data.
E
Detailed explaination of MPDPS
The primary goal of MPDPS to exploit the temporal continuity of the core sequence to smoothly propagate guidance from limited observations across the entire sequence. For simplicity, consider the setup Fig. E where we aim to generate the core tensors three target timesteps: 4,, t', and 1++1, denoted as Wt,. Wr, and W 11j1 respectively. In this example, observations are only avail- able at l and 44+11 (i.e., observation dimesteps),denoted as Or and Ot, Standard DPS methods cannot provide gradient guidance for generating the intermediate core WH, due to the lack of direct observations at t'.
MPDPS addresses this limitation by propagate the information of On to Wt: Wy, W Wiys and the information of Oh.,, W., We, We+: which means each observation set will contribute the generation of all target core sequences. Next, we demonstrate how this can be achieved.
Without loss of generality, we take Oi, obtain gradient guidance for the three cores:
example. At diffusion step s,
O twice to
i) Gradient for W This follows standard DPS. We directly compute the likelihood gradient with respect to W as V W, log (O,, ),,, as illustrated by the green & red arrows at ty in Fig. Dia).
ii) Gradient for W and W te41
Compute denoised cores at t' and t-+1: first W and W Whts to estimate the corrsponding denoised cores W Do(W;) and w Wayps Da(Wi+) via the pre- trained denoiser (.).)
Estimate pseudo cores at t via GPR: With the temporal continuity assumption, we use Gaussian Process Regression (GPR) to estimate the pseudo denoised core W", = 0 en W GPR(D, Do(Wi which can be viewed as a regression- GPR(MV
based prediction using two denoised cores.
Gradient computation: With pseudo core W. compute additional likelihood of O,,- Then we can then compute the gradient of the likelihood respect to W and W as they directly determine W via Da and GPR. We further derive closed-form solution in Appendix D for gradients with respect W and W Whe, (i.e., Vw; lo, ), which can be efficiently computed. The results are shown in Eq. [0
The process is illustrated by the green arrows pointing to O2 in Fig. (bb).
Similarly, information from Or,,, is propagated to Wr. We and W in the same manner, and we sum the gradients from all observations to obtain the final gradient guidance for each core, as shown in Eq. P
In this way, each observation contributes to the generation of all target timesteps. In other words, every target coreregardless of whether has a corresponding observationreceeivs information from all observed timesteps. To sample the entire core sequence, we simply add the gradients from all observations to the posterior, as shown in Eq. [2] This mechanism encourages smooth and coherent generation across the entire sequence.
26

---

Implementation details
All the methods are implemented with PyTorch [+2]] and trained using Adam + optimizer with the learning rate tuned from (5e-*, 1e- 5e
le 1e_
For DEMOTE, we used two hidden layers for both the reaction process and entry value prediction, with the layer width chosen from {128, 256, 512]. For LRTFR, we used two hidden layers with layer width chosen from {128, 256. 512} to parameterize the latent function of each mode. We varied R from {16, 32} for all baselines. For Senseiver, we used 128 channels and decoder, a sequence size of 256 for the Qin array, and set the size of the linear layers in the encoder and decoder 128. For CoNFiLD, we used Conditional Neural Field module with a latent dimension of 256 for the Ocean Sound Speed and Active Matter datasets, and 1024 for the Supernova Explosion dataset. The diffusion model module was configured with 100 sampling steps. For our method, we first apply functional Tucker model to decompose the tensor into factor functions and a core sequence. Each factor function is parameterized by three-layer MLP, where each layer contains 1024 neurons and uses the sine activation function. The core sizes are set to 32 32 x 32, 3 12, and 48 x 48 for the Supemova Explosion, Ocean Sound Speed, and Active Matter datasets, respectively. These hyperparameters are carefully selected to achieve optimal performance.
both the encoder
F.1 Defination of Variance-scaled Root Mean Squared Error
Let [0: and { denote the predicted and ground-truth entry, respectively, Assume that there are N points in total. The seriance-scales Root Mean Squared Error (VRMSE) is defined as
N
0s
y1)
VRMSE
1=1
(41)
Σ
y3 7)
N
where ý the mean of all points.
F2 Guided Sampling Details
Supernova Explosion
Ocean Sound Speed
Active Matter
{SDIFT
10
3 10
1 10
SCaNFLLD x 10-1
x 10-1
1.2 x 10-1
Table 3: Weights assigned to posterior guidance by CoNFiLD and SDIFT.
For experiments conducted on three datasets with sparse observations, we use the weights Ç in Tab.
G Additional experiment results
G.1 Unconditional generation
In this subsection, we present unconditional generation results on three datasets of SDIFT, demon- strating its ability generate physical fields from irregular and sparse training data. The uncondi- tional generation results on Supernova Explosion are shown in Fig. E The unconditional generation results on Ocean Sound Speed are shown in Fig. The unconditional generation results on Active Matter are shown in Fig. & One can see that the unconditional generations are of high quality and smooth across frames, demonstrating the effectiveness SDIFT in generating full-field physical d dynamics from irregular sparse observations.
27

---

frame #0
frame #2
frame #4
frame #6
frame #8
frame #10
frame #12
frame #14
Figure 6: Unconditional generation results on Supernova Explosion by SDIFT.
28

---

frame frame frame
frame #9
frame frame frame #12
frame #21
Figure 7: Unconditional generation results on Ocean Sound Speed by SDIFT.
G.2 More reconstruction results
More visual results of three datasets conducted on observation setting are illustrated in Fig.
More visual results of three datasets conducted observation setting 2 illustrated in Fig.
29

---

frame #0
frame #3
frame #6
frame #9
frame #12
frame #15
frame #18
frame #21
Figure 8: Unconditional generation results Activate Matter by SDIFT.
30

---

SDIFT
SDIFT
(MPDPS)
(DPS) VRMSE:0.262
Senseiver VRM5E:0.4
CoNFILD
LRTFR VRMSE:0.462
Ground Truth Observation
VRMSE:0.233
VRM5E:0.57
frame frame
frame
frame frame frame
Figure Visual reconstruction results Supernova Explosion dynamics under observation setting 1 with p 1%.
SDIFT VANSE:00009
Ground Truth
Observation
VAMSSE00.174
VSNMSE::.200
CONFIID L. URMSE
VRMSEID
frane #0
Trame #4
frame #e
frame #12
frame #16
Fran #20
Figure 10: Visual reconstruction results of Ocean Sound Speed dynamics under observation setting
with p = 3%.
31

---

Ground Truth Observation
VRMSE:0.122 VRMSE:0.232
VRMSE:0.48 VRMSE:0.216
frame
frame frame #10 frame frame #4
Figure 11: Visual reconstruction results of Active Matter dynamics under observation setting 1 with = 1%.
SDIFT (MPDPS) VRMSE:0.278
SDIFT (DPS) VRMSE:0.599
CONFILD VRMSE:0.634
LRTFR VRMSE:1.016
Ground Truth Observation
frame #o
frame #3
frame #6
frame #9
frame #15
Figure 12: Visual reconstruction results of Supernova Explosion dynamics under observation set-
ting with p 3%.
32

---

SDIFT VRMSE:00065
SDIFT VRMSE:0.125
Ground Truth
Observation
VRMSE::0.24
VAMSE:0.154
frame frame frame frame
frame #12
frame #15
frame
Figure 13: Visual reconstruction results of Ocean Sound Speed dynamics under observation setting 2 with 3%.
33

---

SDIFT (MPDPS)
SDIFT (DPS) VRM5E:0.616
CoNFILD
LRTFR 18MSE:1.111
Ground Truth Observation
VRMSE:0.345
VRMSE:0.9
frame #0
frame #3
frame #6
frame #9
frame #12
frame #15
frame #18
frame #21
Figure 14: Visual reconstruction results of Active Matter dynamics under observation setting with م = 3%. Since DPS does not provide guidance timesteps without observations (i.e., frame # 3,9,15,21), SDIFT generates the corresponding physical fields randomly. In contrast, MPDPS effectively overcomes this limitation and produces smooth reconstructions. This phenomenon high- lights the significance of the proposed MPDPS.
34

---

G.3 Analysis on the results of CoNFiLD
CoNFiLD [S] is closely related to our approach. It first uses conditional neural fields to learn latent vectors for off-grid physical field data over time. Next, it simple concatenates the T latent vectors extracted from a record into latent matrix and treats this matrix as an image, applying a standard diffusion model to learn its distribution. We present dimensionality reduction visualizations of the latent vectors and core tensors extracted from Acfive Matter dataset using CNF and our FTM, re- spectively, in Fig. (b)) We use points with the same color hue to represent representations from the same record, and vary the color intensity indicate different timesteps. We plot 9 records, with each records containing 24 timesteps. One can see that both kinds of representation show strong time continuity.
CoNFiLD use traditional UNet to learn the distribution of latent matrices (i.e., the sequence of scatter points in Fig. a))),, which is hard to capture complex temporal dynamics (due to the inductive bias of EE Our experiments demonstrate that, although CNF produces accurate reconstructions from its latent representations, the resulting latent matrices are difficult to model with standard diffusion approachesespecially when the physical field changes rapidly, as in the Active Matter datasetleading to poor posterior sampling performance. On the Ocean Sound Speed dataset, where the field varies much more slowly and smoothly, the diffusion model performs significantly better.
In contrast, we propose a temporally augmented U-Net that learns the distribution of each core point (i.e., the individual points in Fig. [S(b) while capturing temporal correlations through the Conv ID module. This approach significantly simplifies modeling the full latent distribution, whether it changes rapidly or slowly. Furthermore, our approach can flexibly generate sequences of arbitrary length with arbitrary time intervals, whereas CoNFiL. cannot.
-2
Component
(a) Latents from CNF.
(b) Cores from FTM.
Figure 15: Illustration of dimensionality reduction of the latent vectors and core tensors extracted from Active Malter using CNF and FTM, receptively.
G.4 Robustness against noise
We demonstrate the robustness of our method against different levels and different types of noise in T
G.5 Ablation study using GP noise the diffusion source
We conduct ablation study the of GP noise the diffusion source by replacing it with i.i.d. Gaussian noise, using the Active Matter dataset as an example.
Fig. 16 shows unconditional generations from our method when i.i.d. Gaussian noise is used as the diffusion source. Compared to Fig. this replacement significantly disrupts temporal continuity and degrades generation quality.
35

---

SDIFT w/ DPS
SDIFT w/ MPDPS
Noise configurations
1%
P 1%
Gaussian noise =0.1)
0.207 0.066
0.156 + 0.052
Gaussian noise (ar = 0.3)
0.210 0.069
0.164 = 0.049
Laplacian noise (a = 00.1)
0.200 0.075
0.170 + 0.054
Laplacian noise (a 0.3)
0.224 1 0.054
0.177 + 0.051
Poisson noise (a 0.1)
0.186 + 0.081
0.168 + 0.048
Poisson noise (o 0.3)
0.214 + 0.070
0.171 = 0.053
Table 4: VRMSE of reconstruction of our proposed method on Ocean Sound Speed datasets over varying noise with observation setting and p 1%.
observation setting
observation setting 1%
VRMSE
1%
noise(ty = 50)
0.215 0.068
0.296 0.096
GP noise(r = 100)
0.222 + 0.077
0.304 + 0.104
GP noise(7 = 200)
0.219 + 0.084
0.298 + 0.084
i.idd Gaussian noise
0.257 1 0.079
0.393 1 0.109
Table 5: VRMSE on Active Malter dataset over different noise source using SDIFT guided by MPDPS.
We also report the VRMSE of reconstruction results in Table B. Replacing GP noise with i.idd Gaussian noise as the diffusion source markedly worsens performance. In contrast, GP noise yields consistently low VRMSE and robust variations in the kernel hyperparameter 7, further demon- strating its suitability as diffusion source in our settings.
H
Limitations
current limitation of our work is the lack explicit incorporation of physical laws into the mod- eling process. In addition, the introduced Gaussian Process prior relies on the choice of kernel functions; this work, we only use the simple RBF kernel as an example and do not explore other options in detail.
In future work, we plan to integrate SDIFT with domain sppciiff physical knowledge to enable more accurate long-range and wide-area area reconstructions of physical fields. We also intend to investigate a broader range of kernel functions enhance modeling flexibility and performance.
I Impact Statement
This paper focuses advancing generative modeling techniques for physical fields. We mindful of the broader ethical implications associated with technological progress in this field. Although immediate societal impacts may not be evident, recognize the importance of maintaining ongoing vigilance regarding the ethical use these advancements. It is crucial to continuously evaluate and address potential implications to ensure responsible development and application in diverse scenarios.
36

---

frame #0
frame #3
frame #6
frame #9
frame #12
frame #15
frame #18
frame #21
Figure 16: Ablation study: Unconditional generation results on Activate Matter by SDIFT using i.i.d
Gaussian noise the diffusion source. Note that this replacement significantly disrupts temporal continuity and degrades generation quality.
37


Reviews:
[{"review_id": "wq5dmaaKNf", "reviewer": "Reviewer_wHTh", "summary": "The authors present Sequential Diffusion in Functional Tucker Space (SDIFT). They deal with the problem of posterior sampling of a continuous-time real-vector valued sequence (like solutions of a PDE). Motivated by latent space diffusion, they do posterior sampling given partial observations of the whole continuous time signal by doing diffusion modeling and posterior sampling in the latent space of a functional tucker tensor decomposition. In the work they deal with two main challenges (1) define the latent space according to the tucker decomposition in order to handle sparse/off-time-grid observations and (2) perform inference using likewise incomplete observations to generate a full continuous time field.", "questions": "- in section 3 124, for the diffusion prior pre-training phase, what's the definition of homogenous in \"B batches of homogeneous dynamics\" mean?\n\n- in section 3.1, line 166, in equation 8 , do you need a coefficient in front of the TV term in practice? If so, how do you choose it? \n\n- section 3.2, line 192, equation 9, there is a denoiser-matching objective to train the diffusion model for fixed W.  Have you tried training (8) and (9) simultaneously rather than in pieces. Any +/-'s to report?\n\n- does temporally augmented Unet just mean that the Unet gets the noised $(W_t)^{s}$ (W for data time t at diffusion time s) as well the times $t=[t_1,...t_M]$ themselves? If not, what does it mean?\n\n- *Main question*: What is proved that the specific message passing algorithm introduced, which uses $\\nabla_{W^s_{\\setminus\nt_\\ell}} \\log p(O_{t_\\ell} | W^s_{\\setminus\nt_\\ell})$ ? I get intuitively that we are passing some info from the observed to the rest, but abstracting a bit, how can I see more easily that updating the whole $W_t$ according to these gradients produces a sample of $W_t^{\\text{all}} | W_t^{\\text{obs}}$ ?", "limitations": "Yes.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 4, "strengths_and_weaknesses": "Strengths\n\n- precise mathematical formalization of the problem at hand\n- moving between typical VAE latent space to something more structured that is used in the applied math / numerical community, but mixing with strengths of ML + diffusion. The time-varying core tensors $W_{t_k}$ of the tucker decomposition are taken as the latent representation so that for data spatial index $i_k$ (very loosely) we have$y_{i,t} = w_t f^k_\\theta(i_k)$ for learnable $f^k_\\theta$ and learnable $w_t$ for mode $k$ . The cores $w_t$ share info across the $K$ modes' latent functions  \n- the latent space $W(t)$ is still at time varying function, so they use functional diffusion models to model it (functional diffusion model has GP noise)\n- the diffusion pre-training and posterior inference algorithms are separated into two clean pieces (where the first learning piece itself has two parts: learn the decomposition, learn the diffusion)\n- novel message passing algorithm to perform the inference\n- convincing experiments on real scientific tasks (1) supernova explosion temperature evolution (2) pacific ocean sound speed (3) active particles in simulated fluid with comprehensive description/links for more details to all data details.\n\nWeaknesses\n\nSee questions.", "was_revised": true, "final_justification": "My positive opinion of the work remains unchanged.", "rebuttal": "Thank you for your strong support!  Here are our responses. C: comments; R: response.\n\n>C1:in section 3 124, for the diffusion prior pre-training phase, what's the definition of homogenous in \"B batches of homogeneous dynamics\" mean?\n\nR1: Thank you for your question. In generative modeling, people assume that the training data is extracted from a specific distribution. Here, we also assume the training data is extracted from a specific distribution of  dynamics. The homogeneous dynamics refer to the same type of dynamics, such as the same PDE or ODE, but with different initial conditions. \n\n>C2:in section 3.1, line 166, in equation 8 , do you need a coefficient in front of the TV term in practice? If so, how do you choose it?\n\nR2: Good point! We add a coefficient in front of the TV term. We have done cross-validation experiments to choose the coefficient and we find that $10^{-6}, 10^{-7}, 10^{-8}$ all work. We will clarify it in the latest version.\n\n\n>C3:section 3.2, line 192, equation 9, there is a denoiser-matching objective to train the diffusion model for fixed W. Have you tried training (8) and (9) simultaneously rather than in pieces. Any +/-'s to report?\n\nR3:Good question. Theoretically, (8) and (9) can be trained simultaneously. However, the objectives of FTM and GPSD differ significantly:\n\n- **FTM** aims to learn a high‑fidelity mapping between the data domain and latent space, minimizing reconstruction error.  \n- **GPSD** aims to model the distribution of latent‑space dynamics over time.\n\nJointly optimizing these distinct objectives leads to slower training and reduced stability.\n\nTo address this, we adopt a two‑stage training pipeline：first training the FTM, then freezing it and training the GPSD. This strategy is widely used in the computer vision literature [A1], which has been shown to yield superior generation quality, faster sampling, and more stable convergence compared to end‑to‑end training.\n\n\n\n\n[A1] Rombach, Robin, et al. \"High-resolution image synthesis with latent diffusion models.\" Proceedings of the IEEE/CVF conference on computer vision and pattern recognition. 2022.\n\n>C4:does temporally augmented Unet just mean that the Unet gets the noised \n ($\\mathbf{W}\\_{t}^{s}$ for data time t at diffusion time s) as well the times $t=[t\\_1,\\cdots,t\\_M]$ themselves? If not, what does it mean?\n\n R4: The answer is yes. Apart from that, we also  use Conv1D layers to capture temporal correlations among $M$ frames. The detailed archetecture is provided in Appendix B.\n\n\n>C5:Main question: What is proved that the specific message passing algorithm introduced, which uses $\\nabla\\_{W\\_{\\setminus t\\_l}^s}\\log p(O\\_{t\\_l}|W\\_{\\setminus t\\_l}^s)$ . I get intuitively that we are passing some info from the observed to the rest, but abstracting a bit, how can I see more easily that updating the whole $\\mathbf{W}\\_t$ according to these gradients produces a sample of $\\mathbf{W}\\_t^{\\text{all}}|\\mathbf{W}\\_t^{\\text{obs}}$.\n\n\nR5: Thank you for your question! We want to highlight that the primary goal of MPDPS is to exploit the temporal continuity of the core sequence to **smoothly propagate guidance from limited observations** across the entire sequence.\n\nFor clarity, consider the setup in Fig. 2, where we aim to generate the core tensors at three target timesteps: $t\\_l$, $t'$, and $t\\_{l+1}$, denoted as **$\\mathcal{W}\\_{t\\_l}$**, **$\\mathcal{W}\\_{t'}$**, and **$\\mathcal{W}\\_{t\\_{l+1}}$**, respectively. In this example, observations are only available at $t\\_l$ and $t\\_{l+1}$ (observation timesteps),denoted as $\\mathcal{O}\\_{t\\_l}$ and $\\mathcal{O}\\_{t\\_{l+1}}$. Standard DPS methods cannot provide gradient guidance for generating the intermediate core **$\\mathcal{W}\\_{t'}$**, due to the lack of direct observations at $t'$.\n\n**MPDPS addresses this limitation by propagate the information of $\\mathcal{O}\\_{t\\_l}$ to {**$\\mathcal{W}\\_{t\\_l}, \\mathcal{W}\\_{t'},\\mathcal{W}\\_{t\\_{l+1}}$**} and the information of $\\mathcal{O}\\_{t\\_{l+1}}$ to {**$\\mathcal{W}\\_{t\\_l}, \\mathcal{W}\\_{t'},\\mathcal{W}\\_{t\\_{l+1}}$**},  which means each observation set will contribute to the generation of all target sequences.**\n\n\n\nWithout loss of generality, we take $\\mathcal{O}\\_{t\\_l}$ as an example. At diffusion step $s$, we use $\\mathcal{O}\\_{t\\_l}$ *twice* to obtain gradient guidance for the three cores:\n\n\n- **First use to compute gradient for $\\mathcal{W}\\_{t\\_l}^s$**:  This follows standard DPS. We directly compute the likelihood gradient with respect to $\\mathcal{W}\\_{t\\_l}^s$ as  $\\nabla\\_{\\mathcal{W}\\_{t\\_l}^s} \\log p(\\mathcal{O}\\_{t\\_l} | \\mathcal{W}\\_{t\\_l}^s)$, as illustrated by the green& red arrows at $t\\_l$ in Fig.2(a).\n\n- **Second use to compute gradients for $\\mathcal{W}\\_{t'}^s$ and $\\mathcal{W}\\_{t\\_{l+1}}^s$**:  \nIt includes three steps:\n    1.  **Compute denoised cores at $t'$ and $t\\_{l+1}$**: we first use $\\mathcal{W}\\_{t'}^s$ and $\\mathcal{W}\\_{t\\_{l+1}}^s$ to estimate the corrsponding denoised cores $\\hat{\\mathcal{W}}\\_{t'}^0 = D\\_{\\theta}(\\mathcal{W}\\_{t'}^s)$ and $\\hat{\\mathcal{W}}\\_{t\\_{l+1}}^0 = D\\_{\\theta}(\\mathcal{W}\\_{t+1}^s)$ via the pretrained denoiser $D\\_{\\theta}(\\cdot)$.\n\n    2. **Estimate pseudo cores at $t\\_l$ via GPR**:  With the temporal continuity assumption, we use Gaussian Process Regression (GPR) to estimate the pseudo denoised core  $\\hat{\\mathcal{W}}\\_{t\\_l}^0 = GPR (\\hat{\\mathcal{W}}\\_{t'}^0, \\hat{\\mathcal{W}}\\_{t\\_{l+1}}^0) = GPR(D\\_{\\theta}(\\mathcal{W}\\_{t'}^s), D\\_{\\theta}(\\mathcal{W}\\_{t+1}^s))$ , which can be viewed as a regression-based prediction using two denoised cores. \n    3. **Gradient computation**: With pseudo core $\\hat{\\mathcal{W}}\\_{t\\_l}^0$ $\\quad$, we compute an additional likelihood of $\\mathcal{O}\\_{t\\_l}$. Then we can then compute the gradient of the likelihood respect to ${\\mathcal{W}}\\_{t'}^s$ and ${\\mathcal{W}}\\_{t\\_{l+1}}^s$, as they directly determine $\\hat{\\mathcal{W}}\\_{t\\_l}^0$ via $D\\_{\\theta}$ and GPR. We further derive a **closed-form solution** in Appendix C for gradients with respect to $\\mathcal{W}\\_{t'}^s$ and $\\mathcal{W}\\_{t\\_{l+1}}^s$ (i.e., $\\nabla\\_{W\\_{\\setminus t\\_l}^s}\\log p(O\\_{t\\_l}|W\\_{\\setminus t\\_l}^s)$ where  $W\\_{\\setminus t\\_l}^s=${$\\mathcal{W}\\_{t'}^s,\\mathcal{W}\\_{t\\_{l+1}}^s$}), which can be efficiently computed. The results are shown in Eq.(10)(11).\n    The process is illustrated by the green arrows pointing to $\\mathcal{O}\\_{t\\_l}$ in Fig. 2(b). \n\n\n\nSimilarly, information from $\\mathcal{O}\\_{t\\_{l+1}}$ is propagated to **$\\mathcal{W}\\_{t\\_l}$** 、**$\\mathcal{W}\\_{t'}$**, and **$\\mathcal{W}\\_{t\\_{l+1}}$** in the same manner, and we sum the gradients from all observations to obtain the final gradient guidance for each core, as shown in Eq. (12).\n\nIn this way, each observation contributes to the generation of all target timesteps. In other words, every target core—regardless of whether it has a corresponding observation—receives information from all observed timesteps. To sample the entire core sequence, we simply add the gradients from all observations to the posterior, as shown in Eq. (12). This mechanism encourages smooth and coherent generation across the entire sequence.\n\n\nWe hope this explanation clarifies how MPDPS updates the entire core sequence. We will include this in the latest version."}, {"review_id": "xXsEfOmfWQ", "reviewer": "Reviewer_dp1M", "summary": "This paper presents SDIFT (Sequential Diffusion in Functional Tucker space), a new generative framework for reconstructing multidimensional physical dynamics from irregular and sparse spatiotemporal observations. The key innovation is the use of a functional Tucker model for latent space representation, combined with a Gaussian Process-based sequential diffusion to generate core sequences, and a Message-Passing Posterior Sampling mechanism (MPDPS) for effective conditional sequence generation. Theoretical foundation is added via a universal approximation guarantee for the functional Tucker model. The method is evaluated on three diverse real-world physical datasets (astronomical, oceanographic, and molecular), showing significant improvements in accuracy and computational efficiency over state-of-the-art baseline approaches.", "questions": "See weekness.", "limitations": "See weekness.", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "**Strengths:**\n\n- The experimental evaluation is thorough, spanning three distinctly different domains (supernova, ocean sound speed, active matter) and comparing against a diverse set of relevant baselines. The method demonstrates consistent quantitative improvements, as detailed in Table 1 (Section 5): SDIFT with MPDPS achieves lowest VRMSE scores across datasets and observation scenarios, often with large gaps to other methods.\n\n- The figures provide clear qualitative evidence: Figure 3 and Figure 4 demonstrate that SDIFT with MPDPS yields visually superior reconstructions at both dense and missing observation times, compared to competing approaches. In particular, Figure 4 highlights the advantage of the message-passing mechanism in interpolating missing temporal slices.\n\n**Weaknesses:**\n\n- The comparison with baselines is not comprehensive. Although the authors compare parameter size and runtime in Table 2, this comparison is limited to CoNFiLD only and does not include other baselines. I am particularly curious about how the proposed method performs against the other reported results.\n\n- The reviewer is confused about the content in Section 3.1. In this section, the authors propose the Functional Tucker Model (FTM) as a universal latent representer. For Equation (6), let us consider a two-dimensional scenario and even ignore the temporal index $t$, assuming $y(i_x, i_y, t) = e^{i_x \\cdot i_y}$. The reviewer would like the authors to provide finite-dimensional solutions for $f^1_{\\theta_1}(i_x)$ and $f^2_{\\theta_2}(i_y)$ in this example.\n\n- Building upon the previous concern, for the case $y(i_x, i_y, t) = e^{i_x \\cdot i_y}$, the reviewer argues that only an approximate solution can be obtained under finite-dimensional settings. This raises a new issue: in the proof of universal approximation, the authors must account for the case where $R^k \\to \\infty$. Therefore, the reviewer is confused by the last step of Equation (24) on page 15.\n\n- There is a typographical error in the formula on page 14, Equation (18). In the $B_2$ component, $f_{r_1}^{A_1}$ should be $f_{r_1}^{k}$.\n\n- Minor issue: references [28] and [38] refer to the same publication.", "was_revised": true, "final_justification": "I still have some reservation on the novelty of this work (incremental compared with existing tensor learning methods). Hence, I maintain my original rating.", "rebuttal": "We thank the reviewer for the careful review! We address the comments below:( C: comment; R: response)\n\n\n>C1: The comparison with baselines is not comprehensive. Although the authors compare parameter size and runtime in Table 2, this comparison is limited to CoNFiLD only and does not include other baselines. I am particularly curious about how the proposed method performs against the other reported results.\n\nR1: Thank you for the comment. We compared our method only with CoNFiLD because both are generative approaches. The parameter sizes and runtimes for the other methods are reported below.\n\n|     |SDIFT (ours) |CoNFiLD|NONFAT | DEMOTE | LRTFR | Senseiver|\n|:-----:|:-----:|:-----:|:-----:|:-----:|:-----:|:-----:|\n| Param. size on Supernova Explosion |26M|23M| N/A| 6.5M |4.2M|9.3M|\n| Samp. time and VRMSE  on Supernova Explosion (1%) |**2.23s(0.283$\\pm$0.026)**|31.3s(0.561$\\pm$0.082)| $>30$min(1.229$\\pm$0.127)| $>30$min(1.285$\\pm$0.102) |31.2s(0.558$\\pm$0.044)|0.83s(0.446±0.041)|\n| Samp. time and VRMSE on Supernova Explosion  (3%)|**5.43s(0.272$\\pm$0.025)**| 44.0s(0.427$\\pm$0.037)| $>30$min(1.197$\\pm$0.204) | $>30$min(1.213$\\pm$0.217) |35.4s(0.429$\\pm$0.043)|1.5s(0.349±0.023)|\n| Param. size on Ocean Sound Speed |15M|10M| N/A| 5.8M |3.2M|8.5M|\n| Samp. time  and VRMSE on Ocean Sound Speed  (1%)|**0.84s(0.146$\\pm$0.046)**|15.9s(0.201$\\pm$0.034)| $>30$min(0.402$\\pm$0.090) | $>30$min(0.358$\\pm$0.127) |27.5s(0.345$\\pm$0.036)|0.036s(0.264±0.037)|\n| Samp. time and VRMSE on Ocean Sound Speed  (3%)| **0.89s(0.108$\\pm$0.043)**|20.3s(0.145$\\pm$0.012)| $>30$min(0.330$\\pm$0.101) | $>30$min(0.314$\\pm$0.086) |28.0s(0.217$\\pm$0.066)|0.04s(0.2005±0.031)|\n| Param. size on on ActiveMatter  |12M|10M| N/A | 4.7M | 3.2M|8.2M\n| Samp. time and VRMSE on Active Matter  (1%)|**1.31s(0.215$\\pm$0.068)**| 27.6s(0.529$\\pm$0.087)| $>30$min(0.921$\\pm$0.457) | $>30$min(0.950$\\pm$0.486) |26.9s(0.302$\\pm$0.104) |0.14s(0.345±0.094)\n| Samp. time and VRMSE on Active Matter  (3%)|**1.42s(0.156$\\pm$0.046)**| 31.5s(0.5075$\\pm$0.830)|$>30$min(0.867$\\pm$0.413)| $>30$min(0.871$\\pm$0.497) | 27.3s(0.258$\\pm$0.022) |0.18s(0.264±0.076)\n\n\nThe parameter sizes of different models vary because they are different types of models and  we fine-tune each model individually to achieve its best performance, taking into account its unique properties.\n\nWe observe that tensor-based methods (NONFAT, DEMOTE, and LRTFR) require significantly more time to produce reconstructions. This is because they directly minimize a loss function that enforces consistency between predicted and observed values, without any pretraining phase. As a result, these methods need extensive iterations to converge and are much slower than pretrained models. Among them, LRTFR is considerably faster than NONFAT and DEMOTE, as the latter two are not scalable to large tensors.\n\n\n\nSenseiver is the most efficient model in terms of runtime, as it feeds the indices to be queried into the network and performs the forward pass only once to produce the reconstructions. However, it does not explicitly model the temporal evolution of the physical field. As a result, its performance is inferior to ours and it fails when no observations are available.\n\n\n>C2: The reviewer is confused about the content in Section 3.1. \n\nR2: Thank you for your question. In your example, $y(i\\_1, i\\_2) = e^{i\\_1 \\cdot i\\_2}$, **we can construct a finite-dimensional FTM solution that exactly represents $y$.** That is $y(i\\_1,i\\_2)= \\text{vec}(\\mathbf{W})^{\\text{T}}(f\\_{\\theta\\_1}^1(i\\_1) \\otimes f\\_{\\theta\\_2}^2(i\\_2))$, where $\\mathbf{W}$ being a scaler 1 and $f\\_{\\theta\\_1}^1(\\cdot)=e^{i\\_1}: \\mathbb{R}^1 \\to \\mathbb{R}^1$,$f\\_{\\theta\\_2}^2(\\cdot)=e^{i\\_2}: \\mathbb{R}^1 \\to \\mathbb{R}^1$    being scaler functions. Then the FTM model becomes $1\\cdot(f\\_{\\theta\\_1}^1(i\\_1) \\cdot f\\_{\\theta\\_2}^2(i\\_2))$. Since both latent functions are parameterized by neural networks, they can accurately approximate the exponential functions involved.\n\n\nFor illustration, let's consider a finite 2-order tensor(matrix) $\\mathbf{Y}$ with size $3 \\times 3$ with each entry $y(i\\_1,i\\_2)=e^{i\\_1,i\\_2}$. Then  $\\mathbf{Y} = $\n\n|     |     |     |\n|:-----:|:-----:|:-----:|\n| y₁₁ | y₁₂ | y₁₃ |\n| y₂₁ | y₂₂ | y₂₃ |\n| y₃₁ | y₃₂ | y₃₃ |\n\n=\n|     |     |     |\n|:-----:|:-----:|:-----:|\n| $e^1$ | $e^2$ | $e^3$ |\n| $e^2$ | $e^4$ | $e^6$ |\n| $e^3$ | $e^6$ | $e^9$ |\n\n\nFTM decomposes the tensor $\\mathbf{Y}$ into a Tucker core and mode-wise latent functions.  \nAssume $R\\_1 = R\\_2 = 1$, then the Tucker core is a learnable matrix $\\mathbf{W} \\in \\mathbb{R}^{1}$, and the latent functions are defined as:\n- Mode-1: $f\\_{\\theta\\_1}^1(\\cdot): \\mathbb{R}^1 \\to \\mathbb{R}^1$  \n- Mode-2: $f\\_{\\theta\\_2}^2(\\cdot): \\mathbb{R}^1 \\to \\mathbb{R}^1$  \n\nHere, $f\\_{\\theta\\_1}^1(\\cdot)$ and $f\\_{\\theta\\_2}^2(\\cdot)$ are neural networks that learn positional embeddings and are parameterized by $\\theta\\_1$ and $\\theta\\_2$.\n\nTo approximate the value at index $(1, 1)$, FTM computes: $  \\text{vec}(\\mathbf{W})^\\top \\left( f\\_{\\theta\\_1}^1(1) \\otimes f\\_{\\theta\\_2}^2(1) \\right) \\approx y\\_{11} = e^{1}$.\n\nOther entries in $\\mathbf{Y}$ are computed similarly.\n\nFTM minimizes the reconstruction error between $\\mathbf{Y}$ and the model output, learning optimal values for $\\mathbf{W}$, $f\\_{\\theta\\_1}^1(\\cdot)$, and $f\\_{\\theta\\_2}^2(\\cdot)$.  It still works even when some entries in $\\mathbf{Y}$ are missing.\nAfter training,  $\\left( \\mathbf{W}, f\\_{\\theta\\_1}^1(\\cdot), f\\_{\\theta\\_2}^2(\\cdot) \\right)$ can be used to represent the original tensor $\\mathbf{Y}$. Therefore, FTM serves as an encoder. \n\n\n\n\n>C3: Confusion on the case $R\\_k \\to \\infty$ and  last step of Equation (24).\n\n\nR3:  Thank you for the question. We would like to clarify that the universal approximation proof is based on **sufficiently large but finite** $R\\_k$, rather than taking $R\\_k \\to \\infty$.\n\nAs shown in [A1], the Tucker model is a universal representer: for sufficiently large $\\{R\\_k\\}$—up to $R\\_k = I\\_k$, where $I\\_k$ is the size of mode $k$—any finite-dimensional tensor can be exactly represented via Tucker decomposition.\n\nFor example, consider the above case where $\\mathbf{Y} \\in \\mathbb{R}^{3\\times 3}$, $I\\_1 = I\\_2 = 3$. If we choose $R\\_1 = R\\_2 = 3$, i.e., $\\mathbf{W} \\in \\mathbb{R}^{3 \\times 3}$, then the model  \n\n$$\\text{vec}(\\mathbf{W})^\\top \\left( f\\_{\\theta\\_1}^1(\\cdot) \\otimes f\\_{\\theta\\_2}^2(\\cdot) \\right)$$\n\ncan accurately represent $\\mathbf{Y}$.\nTherefore, $R\\_k$ is a finite and bounded value, and the expression in Eq. (24) consists of finite summations. \n\nMoreover, natural physical fields typically exhibit low-rank structure, as discussed in [20,21], implying that the effective $R_k$ values are often small in practice.\n\n\n\n[A1] N. D. Sidiropoulos, L. De Lathauwer, X. Fu, K. Huang, E. E. Papalexakis, and C. Faloutsos, “Tensor decomposition for signal processing and machine learning,” IEEE Trans. Signal Process., vol. 65, no. 13, pp. 3551–3582, Jul. 2017.\n\n>C4:There is a typographical error in the formula on page 14, Equation (18)\n\nR4: Thank you for pointing out this error. We will fix it in the latest version.\n\n>C5:Minor issue: references [28] and [38] refer to the same publication.\n\nR5: Thank you for pointing out this issue. We will address it in the latest version."}, {"review_id": "Hd2GRqdPdR", "reviewer": "Reviewer_uVzD", "summary": "This paper tackles an important and practically relevant problem—reconstructing full-field spatiotemporal physical dynamics from irregular and sparse observations. The proposed SDIFT framework is methodologically innovative and theoretically sound. It builds on the Functional Tucker Model as a continuous latent representation with proven universal approximation capability, and further integrates a Gaussian process–based diffusion mechanism and a novel message-passing posterior sampling strategy. The technical contributions are solid, well-motivated, and clearly described, with thorough empirical validation across diverse physical systems. Overall, the paper offers a compelling and elegant solution to a challenging generative modeling task.", "questions": "1. Since the method does not require pretraining on densely sampled dynamics, I am curious about its behavior under extremely sparse observation conditions. Specifically:\n\n- As the number of observations decreases, is there a threshold below which the model fails to generate meaningful reconstructions due to a lack of information?\n- How does the model perform under non-uniform observation patterns, such as those common in weather-related applications, where sensors are densely located in developed regions but sparse over oceans or remote areas? This setting might challenge the assumption of homogeneity in sampling patterns.\n\n2. In some domain-specific applications, methods have been proposed to perform field reconstruction without pretraining. For example, [1] uses a learnable AI model to represent the evolution process and directly minimizes a loss function that enforces consistency between the predicted and observed values. This formulation seems more amenable to incorporating physical constraints or domain priors. Could you clarify the main advantages of the method compared to such approaches, particularly in terms of flexibility, generalizability, or computational efficiency?\n\n[1] Fablet, Ronan, Bertrand Chapron, Lucas Drumetz, Etienne Mémin, Olivier Pannekoucke, and François Rousseau. \"Learning variational data assimilation models and solvers.\" *Journal of Advances in Modeling Earth Systems* 13, no. 10 (2021): e2021MS002572.", "limitations": "Yes, the authors have discussed the limitations.", "rating": 5, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 4, "significance": 4, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\n1. Challenging and important problem setting: The paper tackles the highly non-trivial task of reconstructing full-field physical dynamics from sparse, irregular, and off-grid observations. Unlike many existing approaches that require access to dense spatiotemporal trajectories for pretraining, SDIFT is capable of learning directly from sparse observations, making it more practical and broadly applicable in real-world scenarios.\n2. Theoretical soundness: The proposed framework is supported by a solid theoretical foundation. In particular, the authors rigorously prove the universal approximation property of the Functional Tucker Model (FTM).\n3. Comprehensive experiments: The paper presents extensive experimental validation across three distinct physical systems (supernova explosions, ocean sound speed, and active matter), spanning different spatial and temporal scales. \n\nWeaknesses:\n\nI acknowledge that I am not deeply familiar with the literature on tensor-based representations for physical field modeling, so I may have missed potential limitations in that aspect of the method. From my perspective, the overall design appears reasonable and well-motivated. I have some questions about the framework which are left to the \"Questions\" section.", "was_revised": true, "final_justification": "The paper tackles the highly non-trivial task of reconstructing full-field physical dynamics from sparse, irregular, and off-grid observations, by proposing a framework which supported by a solid theoretical foundation. From my perspective, the method is interesting and the overall design appears reasonable and well-motivated. I acknowledge that I am not deeply familiar with the literature on tensor-based representations for physical field modeling, so I may have missed potential limitations in that aspect of the method and can't really evaluate its novelty.", "rebuttal": "We sincerely appreciate your thoughtful feedback and  recognition of our work. Below, we address the comments (C: comment, R: response):\n\n>C1:I acknowledge that I am not deeply familiar with the literature on tensor-based representations for physical field modeling, so I may have missed potential limitations in that aspect of the method.\n\nR1: Thank you for your comment! We would like to note that using tensors to model multidimensional physical fields is still a new research direction with little prior work. Tensor-based representations have been widely used in image processing. However, to the best of our knowledge, we are the first to apply them to generative modeling of physical fields and our extensive experiments deomstrate their effectiveness.\n\n>C2: As the number of observations decreases, is there a threshold below which the model fails to generate meaningful reconstructions due to a lack of information?\n\nR2:  Thank you for the question. Diffusion models generate data based on two main sources of information: (1) the prior distribution learned during pre-training, and (2) the likelihood distribution derived from observed data. As a generative model, a well-trained diffusion model can always produce meaningful outputs even when there are few or no observations, relying mostly on the learned prior when the likelihood information is limited.\nTo illustrated that,  we plot the unconditional generation results in in Fig. 6,7,8 of Appendix F. One can see that even without observations, our method still generate meaningful trajectories. \n\n\n\n\n>C3:How does the model perform under non-uniform observation patterns, such as those common in weather-related applications, where sensors are densely located in developed regions but sparse over oceans or remote areas? This setting might challenge the assumption of homogeneity in sampling patterns.\n\nR3: Thank you for the great question! Following your suggestion, we equally divide the physical field into two regions. Assuming an observation rate of 1% at each timestep, we simulate non-uniform observation patterns by allocating **60%** of the observation points to one region and **40%** to the other. We also test a more imbalanced case, with **80%** of the observations in one region and **20%** in the other. These experiments are conducted on a single sample from each dataset. We report the reconstruction VRMSE below:\n\n| Dataset | Uniform Pattern | Non-uniform Pattern (60%/40%) | Non-uniform Pattern (80%/20%) |\n|:-------:|:----------------:|:------------------------------:|:------------------------------:|\n| Supernova Explosion ($\\rho = $1%) | 0.265 | 0.271 | 0.277 |\n| Ocean Sound Speed ($\\rho = $1%)   | 0.137 | 0.139 | 0.141 |\n| Active Matter ($\\rho = $1%)       | 0.114 | 0.115 | 0.126 |\n\nWe find that our method remains robust under non-uniform observation patterns in different datasets, highlighting its practical usefulness. We will supplement it  in the latest version.\n\n>C4:  Could you clarify the main advantages of the method compared to such approaches, particularly in terms of flexibility, generalizability, or computational efficiency?\n\nR4: We appreciate the reviewer’s insightful comment regarding the comparison with domain-specific methods such as [1], which perform field reconstruction without pretraining. Below, we highlight the advantages of our **SDIFT** framework in terms of **flexibility**, **generalizability**, and **computational efficiency**:\n\n|      **Methods**        | **Flexibility**                                                                                          | **Generalizability**                                                                                  | **Computational Efficiency**                                                                                      |\n|--------------|----------------------------------------------------------------------------------------------------------|---------------------------------------------------------------------------------------------------------|--------------------------------------------------------------------------------------------------------------------|\n| **SDIFT**    | Handles irregular and sparse observations across multiple scales during both training and testing.       | Adapts to arbitrary observation patterns during sampling.                                     | Compresses high-dimensional fields into a low-dimensional latent space. With fixed pre-trained latent functions, the likelihood model is linear, and the gradient-guided sampling is highly efficient. |\n| **Model in [1]** | Relies on predefined dynamical models (i.e., PDE/ODE model) and structured observations.                                    | Limited to low-dimensional chaotic systems with fixed observation schemes.                              | Operates in the original data space using extensive iterative gradient descent, which is costly for large systems. |\n\n\n\n- **SDIFT is able to handle irregular, off-grid observations over continuous spatiotemporal domains.** Leveraging the FTM with the universal approximation property, SDIFT flexibly represents high-dimensional fields into latent space without requiring explicit ODE/PDE formulations. Then, in latent space, SDIFT use GPSD to flexibly model temporally coherent core sequences at arbitrary timesteps. However,  [1] relies on physics-informed architectures with predefined dynamics and structured observations (e.g., periodic sampling). \n\n\n\n- **SDIFT generalizes across diverse physical systems and observation scenarios** through two key components: FTM and MPDPS. FTM constructs high-quality mappings between irregular multidimensional physical data and a lower-dimensional latent space. MPDPS effectively propagates sparse observation information across the entire core sequence, ensuring robust performance with irregular, sparse, or incomplete data. In contrast,  [1] relies on gradient-based solvers that require structured observations and degrade under irregularities.\n\n\n- **SDIFT achieves high sampling efficiency** by operating in a low-dimensional latent space. The FTM inherents a linear structure between latent factor functions and core tensors, resulting in a linear likelihood model. This design makes SDIFT scalable to large, high-dimensional systems. In contrast, [1] requires extensive iterative gradient computations over full state vectors, which becomes computationally prohibitive at scale.\n\n\n- **Regarding physical constraints**:  We agree that [1] enforces physics through explicit modeling, but this comes at the cost of flexibility and generalizability. In contrast, SDIFT implicitly captures physical regularities through FTM’s low-rank spatial structure, spatial smoothness, and GP-based temporal continuity. This design preserves flexibility while allowing the future incorporation of domain-specific priors—striking a balance between physical fidelity and adaptability to under-constrained, sparse data."}, {"review_id": "RPgjIIuENq", "reviewer": "Reviewer_BZPv", "summary": "The paper proposes a latent diffusion model for spatiotemporal data, where the latent representation of each temporal snapshot follows a Tucker decomposition, and diffusion learns the distribution of Tucker factors. The paper proposes message-passing to denoise temporally consistent frames. The method works very well on interpolating frames from very sparse data.", "questions": "- The role of modes could be clarified. The paper starts with talking about K modes, which I think are just the spatial coordinates (ie. x,y,z in 3D data). Do we ever have K>3?\n- Is time one of the modes? Why or why not?\n- What does “solving” eq 6 mean in line 149? One needs here both the W and f’s: are both optimised? How do you optimise? Having an algorithm box would be helpful.\n- The remark about VAE is rather strange: I don’t see anything variational, stochastic or Bayesian around eq 6.\n- Line 163 claims interpretable cores. I don’t see any demonstration of this: how are the cores interpretable?\n- What is “alternating-direction strategy”?\n- It seems that one only learns one scalar function per dimension (line 170). This feels quite limited: I wonder what the f-core is then learning or representing? What do the f-cores look like in practise?\n- Is learning of W and f identifiable? One would expect that some rotational invariances would exist. Is their learning sensitive to initialisation? In line 176 the paper assumes that W_i equals W(t_i), but I don’t think you can state this unless the encoding of W_i is deterministic, which requires identifiability and invariance to initial values. In practise the encoding seems stochastic, and one would then have some stochasticity in the W_i’s you get. I also wonder what “drawn from a continuous function” means. We draw samples from a distribution, but evaluate a function. Is there some stochastic process interpretation here? Can you discuss?\n- In line 178 the W(t) is remarked as distribution. Why is it a distribution? \n- Can you clarify the roles of s and t in eq 9. If t is an observation time, then why not also integrate over observation locations?\n- I’m a bit confused what \\Eps or \\W look like: what are their dimensions? Line 184 says that \\Eps is only a function of time, but it is added to \\W which only contains spatial stuff. Is the idea that Eps-set is kept fixed? Or is there some kind of Eps-minibatching?\n- Line 200 says that we can sample using eq 9 model for any (i,t). Does eq 9 contain i?\n- In line 204 it is stated that observations are a subset of target times. Why do we want to predict target times if we already observe them? Can you clarify what is test and train?\n- I can’t follow eqs 10-12\n- How does the train/test split work in experiments? Do you do interpolation or forecasting? How does this relate to the Tar/obs splitting? It seems that half of target is obs, but 95% of frames are train and 5% are test. I’m not sure I understand what does target/test/obs/train mean.\n- How do you use the model at training and at testing stage?", "limitations": "No issues", "rating": 3, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 1, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Originality good. The paper combines existing ideas, but the use of Tucker as diffusion latent representation is innovative and a good match for the domain problem. \n\nSignificance mixed. The scope of the paper is physics-based, low-resolution spatiotemporal dynamics, where some frames are missing and remaining ones have 1-3% of the pixels observed. The task is to impute the field. The task is interesting and is a great stress-test challenge for the methods. However, the task is also very specific, and the paper does not demonstrate why it’s a significant or realistic application. I'm not sure if the paper demonstrates \"real-world\" usecases. The paper does not discuss if the contributions have more general ML relevance.  \n\nQuality mixed. The method has good and consistent performance. The method is well-motivated and sound. The main issues of quality are in weak clarity. \n\nClarity weak. The method is relatively simple (denoising latent factors), but the notation is overly complex. The paper defines a datapoint as (i_n_m, t_m, y_i_n_m), and most parts of the paper inherit this complexity which makes the paper hard to read and digest. The K-mode notation seems unnecessary since all experiments are in 3D or 2D. The motivation, open problem and setting are vague [is this a PDE paper? What is “off-grid”? Do we interpolate or extrapolate or forecast?], and the paper does not clearly position itself to the literature. The message-passing sections are borderline unreadable to the the complicated notation. The method is a black-box, and little insight is given how the model denoises, or why (visualisations would help). There are no ablations showing how the different terms improve (Tucker vs other latent representations; message-passing vs other temporal consistency guidance methods, etc). The paper needs to dramatically improve it’s notation, and borrow good practises from exemplar diffusion papers.  \n\nOverall the paper is an ok technical report showing a succesful method, but as a scientific paper it needs to demonstrate the ideas and significance in wider ML context, and improve presentation.", "was_revised": true, "final_justification": "The response was informative and resolved my confusions about the work. This is a solid work with great performance and good novel method contribution, but the clarity and wider ML significance are still weaknesses of the work. I'm leaning on borderline rejection, but would not object to acceptance.", "rebuttal": "We thank the reviewer for the careful review and  will improve the clarity in the latest version.  We address the comments below:(C: comment; R: response)\n\nTo aid the reviewer’s understanding, we first claim our paper’s main objective, basic concepts, and experiment setup:\n|     |  | \n|:-----:|:-----|\n|**main objective & real-world applications** | The **main objective** of this paper is to  reconstruct complex time-evolving physical fields from sparse sensor observations. This scientific task is   challenging yet important  across diverse domains, including industry, medicine, and science. Applications range from structural health monitoring, tectonic motion estimation,  climate modeling, and so on [18]. This problem has attracted extensive attention, including in leading journals such as *Nature* [5,6,18]. | \n|**Basic concepts** | **1.Off-grid data** are irregularly sampled, sparse measurements from sensors distributed unevenly in space and time, which cannot be represented as regular grids or tensors. **2.Functional Tucker model for representation**:We use FTM as a universal representor to map each snapshot’s datapoints into a core tensor and $K$ shared latent factor functions. The latent factors are shared across snapshots, while the cores vary. For computational details, pls refer to R2 of Reviewer dp1M.The learned latent functions serve as positional basis functions for each mode, with the core tensor providing their weights. *This formulation decouples the temporal mode (via the core) from spatial or other modes (via the latent factors)*.Thus, we can focus on modeling temporal dynamics by learning the distribution of core sequences using a diffusion model.**3. Latent diffusion model** efficiently maps data (i.e., images or video frames) into a low‑dimensional latent space to perform noise‑based generation and editing,  improving efficiency while preserving generation quality. It generate latents by iteratively denoising a noisy input through a learned reverse process.| \n|**Experiment setup**| Assume we have a total of $B$ physical dynamics, split into a training set with $B_1$ and a test set with $B_2$ dynamics, where $B_1 + B_2 = B$ and the two sets are disjoint. To simulate realistic scenarios where sensor data is inherently sparse and irregular, we randomly mask out 80% of the data in the $B_1$ set, using the remaining 20% to train FTM and GPSD.During testing, we sample only 1% or 3% of the data from the $B_2$ set to reflect practical situations with highly sparse observations. These sparse samples are used to guide reconstruction via gradient updates, using the pre-trained FTM and GPSD. The reconstructed full fields are then compared against the ground-truth $B_2$ dynamics.*In conclusion, our method is able to handle irregular data both in training phase and test phase.* |\n\n>C1: Confusion on more general ML relevance\n\nR1: We propose a new generative model that directly operates on off-grid multidimensional data during training and testing phase, which is broadly applicable to unstructured domains like sensor networks, point clouds, and scientific measurements.\n\n\n\n>C2: Concerns on (i_n_m, t_m, y_i_n_m)  and K-mode notation\n\nR2:We clarify that this work aims to construct a generative model directly from **off-grid** data, which cannot be represented as regular matrices or tensors. Instead, we treat the data as points, following common practice in the literature [12, 31].\n\nThis work builds a generative model from **off-grid** data, treated as point sets rather than regular tensors, following common practice [12, 31].\n\nThe value of $K$ is for notational rigor and is will be greater than 3 in practice—for example, $K = 4$ for 3D spatial + frequency radio map/seismic data, or $K = 5$ for multispectral data with 3D spatial, frequency and observation angular modes. Our experiments focus on 2D and 3D cases for simplicity.\n\n\n\n>C3: Question on the time mode\n\nR3:The temporal mode is not included in $K$. The reason is that the temporal mode usually exhibits much sharper or more complex fluctuations that cannot be effectively captured by a single decoupled  latent factor function [31].Instead, we treat it separately using our proposed GPSD module, which leverages Gaussian processes and a temporally-augmented UNet to model temporal correlations.\n\n>C4:What does “solving” eq 6 mean?What is “alternating-direction strategy”?\n\nR4: Solving Eq. (6) means learning the mapping between data and a low-dimensional latent space by minimizing Eq. (8) using an alternating optimization that updates $\\mathcal{W}$ and $f_{\\theta}$ parameters in turn. \n\n>C5:The remark about VAE is rather strange\n\nR5: We clarify that FTM is similar to VAEs in function—mapping data to a latent space—rather than in specific techniques.\n\n\n>C6: Visualisation for  denoisng  cores.\n\nR6: We will  add visualizations in the latest version.\n\n>C7: Ablations on Tucker vs other latent representations; message-passing vs other temporal consistency guidance methods.\n\nR7:We use FTM as the latent representation to learn a  mapping from irregular multidimensional physical fields to a lower-dimensional space, as other latent representations like VAE cannot  handle such irregular data in our setting.\n\nTo the best of  our knowledge, **we are the first to introduce temporal consistency for diffusion guidance.** However, we can use  common interpolation methods like linear or spline to interpolate gradients for those timesteps without observations.  We conduct experiments on  Observation setting 2 with three datasets. The results are:\n|     | MPDPS | DPS+spline|  DPS+linear| \n|:-----:|:-----:|:-----:|:-----:|\n| Supernova Explosion ($\\rho=$1%) |**0.433$\\pm$0.163** |0.581$\\pm$0.298 |0.621$\\pm$0.227| \n| Ocean Sound Speed ($\\rho=$1%) |  **0.181$\\pm$0.084** | 0.294$\\pm$0.12 | 0.342$\\pm$0.181|\n| Active Matter ($\\rho=$1%) |  **0.296$\\pm$0.096** | 0.487$\\pm$0.210 | 0.512$\\pm$0.233|\n\n One can see that  our MPDPS performs much better. We will supplement these into the latest version.\n\n\n>C8: Missunderstanding on factor function\n\nR8: We kindly refer you to line 146, where we clarify the latent function as **a vector-valued function.** The learned latent functions can be interpreted as positional bases functions for each mode, while the core tensor serves as the weighting coefficients that interact with these bases across the $K$ modes.\n\n>C9: Interpretion of core\n\nR9:The core tensor serves as weighting coefficients for the $K$-mode bases, summarizing the full field at each snapshot. Pls see Sec.6 of [22] for  details of its structural patterns.\n\n>C10:Concern on identifiability, rotational invariance, and sensitivity to initialisation\n\nR10:\n- FTM does not produce identifiable $\\mathcal{W}$ and $f$ due to the freedom of invertible transformations, as you noted, but this is not an issue since our reconstruction task—unlike blind source separation—does not require identifiability. Like deep matrix factorization and VAEs, these transformations leave the reconstruction loss unchanged. Once $f$ is learned and fixed, the rotational ambiguity of $\\mathcal{W}$ is also fixed during diffusion training and sampling.\n\n- For the sensitivity to initialisation, we acknowledge the initialisation is crucial for model training, just like most deep-based models' training. In practice, we train with Kaiming initialisation, AdamW, and (mini‑)batch gradient descent on the convex quadratic decoder of Eq. (6).Across 5 random seeds the VRMSE standard deviation is < $10^{-3}$, showing that the model converges stably regardless of the initial values.We will add this table to the appendix in the latest version.\n\n>C11:question about meaning of “drawn from a continuous function $\\mathcal W(t)$ and stochastic process interpretation”\n\nR11:\n- On the **function** view, we regard the core trajectory as a continuous map $\\mathcal W:\\mathbb R^{+}\\to\\mathbb R^{R_1\\times\\cdots\\times R_K}$. meaning that it can be evaluated at any time point $t$ within the range of interest. The observed cores $\\mathcal W_{t_m}$ are simply samples of this function at discrete time stamps, hence the phrase “drawn”.\n- On the **stochastic‑process** view, in the generative (Stage‑2) part we place an element‑wise Gaussian‑process (GP) prior on $\\mathcal W(t)$ and inject GP noise in the forward diffusion.  The reverse process denoises these stochastic samples.  Thus, in training, we learn the GP prior and the denoiser to model the temporal correlations of the core sequence. This is similar to how standard diffusion models learn the distribution of images or videos in pixel space, but here we do it in a latent function space. In the sampling (Stage‑3) part, we sample the core sequence from the learned GP prior and denoise it to obtain the final core sequence.\n\n>C12:Confusion about the componants of diffusion model (s,t,i)\n\nR12: s:diffusion step; t:time step.  In Stage 1, we use FTM to map all data points at time step $t$ into a Tucker core $\\mathcal{W}_t$. The spatial information is captured by latent factor functions and is decoupled from $\\mathcal{W}_t$. Thus, $\\mathcal{W}_t$ summarizes the full-field information at time $t$ and only the temporal mode needs to be modeled during diffusion training.\n\n\n>C13: Why do we want to predict target times if we already observe them?  \n\n\nR13: Our main task is to reconstruct time-evolving **full fields** from sensor observations. Although the target time points have some sparse observations, they are far from enough for downstream tasks. Hence, completing the entire field is necessary. \n\n\n>C14: I can’t follow eqs 10-12\n\nR14:  For  detailed explanations, pls refer to R5 of Reviewer wHTh.\n\n>C15: Questions on target/test/obs/train\n\nR15: Pls refer to the experiment setup at the begining for the train/test split. In the test phase, the **target time** refers to the timesteps where we aim to reconstruct the full field, while the **observation time** refers to the timesteps where sparse observations are available."}]

Author rebuttal:

Meta-review: This submission received two clearly positive reviews (wHTh, uVzD, both rating 5) and two more skeptical, borderline reviews (dp1M, BZPv, both rating 3). The supportive reviewers praise the method’s practical importance, soundness, and strong empirical results across diverse physical systems. They highlight the novelty of combining a Functional Tucker latent space with Gaussian process–driven sequential diffusion and a message-passing posterior sampling mechanism. Both positive reviewers found the paper well-motivated, technically solid, and convincingly validated.

The more critical reviewers raise concerns primarily about (i) the level of novelty relative to existing tensor-based spatiotemporal methods, (ii) positioning with respect to prior work, (iii) limited discussion of broader ML significance, and (iv) clarity of presentation and notation. While dp1M questions whether the approach is incremental, they acknowledge its technical correctness and strong results, and explicitly state they would not strongly oppose acceptance. BZPv also notes clarity and framing issues, but concedes the method is novel and effective for the stated problem, and would not object to acceptance. These two reviewers are more senior.

The authors provided detailed clarifications and additional experiments. They addressed concerns on baseline coverage, positioned their work relative to prior tensor decomposition methods, and explained the specific advantages over autoregressive approaches and domain-specific PDE solvers. They also added robustness tests for non-uniform observation patterns and clarified the pretraining process. While the skeptical reviewers maintained their original ratings, they indicated no remaining technical objections after rebuttal.

I read the paper and reached conclusions consistent with the reviewers:
 - Strengths: Strong empirical performance on challenging, diverse datasets; technically sound method; interesting integration of latent functional tensor decomposition with generative modeling.
 - Limitations: Novelty is moderate, given prior tensor and GP work; the write-up and notation could be streamlined for clarity; broader ML relevance is not strongly argued.
 
This is an exact borderline case: the paper is well-executed and solves an important, realistic problem, but with moderate novelty and clarity concerns. Given the strong empirical results, the fact that no reviewer is strongly negative, and that the technical correctness is not in dispute, I lean toward giving the benefit of the doubt.
Decision: Accept (poster)