PINFDiT: Energy-Based Physics-Informed Diffusion Transformers for General-purpose Time Series Tasks
Authors: Defu Cao
Wen Ye
Yizhou Zhang
Sam Griesemer
Yan Liu
Venue: ICLR 2026 Poster (2026)
Keywords: Diffusion; Transformer; Time Series; Physics Informed Machine Learning;Physics-Guided Inference in Time Series Diffusion Transformers
Primary area: neurosymbolic & hybrid AI systems (physics-informed, logic & formal reasoning, etc.)
TL;DR: Physics-Guided Inference in Time Series Diffusion Transformers

Abstract: Time series analysis underpins scientific advances. While specialized models have advanced various time series tasks, scientific domains face unique challenges: limited samples with complex physical dynamics, missing observations, multi-resolution sampling, and requirements for physical consistency. With the increasing demands on generative modeling capabilities, we introduce PINFDiT, a diffusion transformer-based model with physics injection during inference. Our approach combines a transformer backbone for capturing temporal dependencies with a comprehensive masking strategy that addresses imperfect data. The diffusion framework enables high-quality sample generation with inherent generative capability. In addition, our model-free physics-guided correction steers generated samples toward physically consistent solutions using calibrated Langevin dynamics, which balances distribution fidelity and physical law adherence without architectural modifications or retraining. Our evaluation demonstrates PINFDiT's effectiveness across multivariate forecasting with imperfect data, physics knowledge incorporation in data-limited scenarios, zero-shot and fine-tuning performance across diverse domains, establishing it as a proto-foundation model that bridges the gap between general-purpose and domain-specific models.

Published as a conference paper at ICLR 2026
PINFDIT: ENERGY-BASED FUSION TRANSFORMERS FOR GENERAL-PURPOSE TIME SERIES TASKS
DIF-
Defu Cao; Wen Ye, Yizhou Zhang, Sam Griesemer, Yan Liu University of Southern California (defucao, yewen, zhangyiz, samgriesemez, yanliu.cs @usc. edu
ABSTRACT
Time series analysis underpins scientific advances. While specialized models have advanced various time series tasks, scientific domains face unique chal- lenges: limited samples with complex physical dynamics, missing observations, multi-resolution sampling, and requirements for physical consistency. With the increasing demands on generative modeling capabilities, we introduce PINFDiT, a diffusion transformer- based model with physics injection during inference. Our approach combines transformer backbone for capturing temporal dependencies with a comprehensive masking strategy that addresses imperfect data. The diffusion framework enables high-quality sample generation with inherent generative capabil- ity. In addition, our model-free seyssics-uiier correction steers generated samples toward physically consistent solutions using calibrated Langevin dynamics, which balances distribution fidelity and physical law adherence without architectural modifications retraining. Our evaluation demonstrates PINFDIT's effectiveness across multivariate forecasting with imperfect data, physics knowledge incorpo- ration in data-limited scenarios, zero-shot and fine-tuning performance across diverse domains, establishing that it bridges the gap between general-purpost and domain-specific models.
INTRODUCTION
Time series analysis serves as a cornerstone methodology underpinning scientific advances across natural science, sustainability research, and healthcare innovation Ye et al. [0224] Cuomo et al. [022]
Burger et al. 2024). While specialized models like TCNs (Franceschi et al. 2019), LSTMs (Siami Namini et al. 2019). GNNs (Wu al., 2020), and Transformers (Woo et al. (024a) have advanced in time series tasks including forecasting, imputation, anomaly detection, and data generation, each task presents unique requirements that difficult to generalize across different applications. Scientific domains introduce additional layers of complexity beyond these general challenges. Experimental data often suffers from limited sample sizes due the prohibitive costs of data collection in fields like high-energy physics, climate science, and biomedical research. Scientific time series frequently exhibit complex, non-stationary dynamics governed by underlying physical laws that simple statis- tical approaches fail to capture. Beyond that, generative modeling capabilities prove essential for scientific time series analysis, offering researchers powerful tools to simulate counterfactual scenarios, supplement sparse datasets, conduct hypothesis testing, and produce physically consistent synthetic data that accurately reflects the complex dynamics underlying natural phenomena.
These challenges are exacerbated by missing observations during critical phenomena, multi-resolution sampling intervals, and the need to incorporate multivariate data streams capturing different aspects of the same underlying process. Specifically, missing values (Kollovieh et al. (023) fragment temporal continuity and create representational gaps that models must learn to bridge or impute, requiring sophisticated handling preserve semantic meaning without introducing artifacts. In addition, mulli-resolution sampling (Niu et al. 2023) introduces inconsistent information density across different variables and time periods, distorting cross- variiale relationships that might otherwise
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reveal important dependencies and forcing models to reconcile information different granularities. Similarly, irregular temporal intervals (Cao et al. 2023a) undermine the equidistant assumptions underlying many architectures, compromising temporal dependency learning and requiring models to adapt to varying time scales simultaneously while maintaining causal consistency. Furthermore, while channel-independen solutions (Nie et al., 2023] Ansari et al.) (024) benefit from temporal relationships, they overlook the correlations between multivariate inputs, resulting in impractical outcomes. Despite these prevalent challenges in scientific applications, current benchmark datasets (Li et al. 2018; Zhouu et al.) 2021] Alexandrov et al. 2020) often present idealized scenarios that inadequately capture the inherent complexities, creating critical gap between controlled evaluation environments and scientific deployment contexts.
Furthermore, scientific applications demand physical consistency and interpretability beyond what black-box models can deliver, as predictions that faithfully adhere to established scientífic principles are required for validation and acceptance (Meng et al., 2022b). For example, in climate science, conservation laws must be preserved despite sparse observations, while in healthcare, physiological constraints like heart rate bounds and circadian rhythms prevent biologically implausible predictions. Incorporating such physics knowledge significantly enhances model performance while reducing computational overhead, especially in data-scarce domains where physical laws provide crucial prior knowledge. Properly modeling such systems requires likelihood-based approaches that quantify uncertainty and provide theoretical guarantees (Simons et al.) (023). However, computing exact likelihoods for complex time series distributions becomes intractable Beal (2003). Traditional simulation-budl based inference methods bypass the need for tractable likelihoods but require prohibitively large numbers of simulations and struggle scale efficiently for time series data, particularly when physical constraints must be maintained across long-term time sequences (Gloeckler et al.) 2024).
In this paper, we introduce PINFDiT, diffusion transformer- based generative model with physics injection during the inference stage to address the aforementioned challenges. The transformer backbone provides natural advantages in capturing complex temporal dependencies across varied time scales, while the diffusion framework enables high-quality sample generation with inherent probabilistic results. To achieve uniform input, we design a comprehensive masking strategy including position, stride, and block masks, which addresses the practical realities of imperfect time series data encountered across domains. Furthermore, we incorporate a model-free physics-guid guided correction that steers the generated samples toward physically consistent solutions during the sampling stage. Different from inailaatio-bbasee inference (Geffner et al., 2023; Xiong et al.) (025). this correction directly addresses likelihood intractability through ELBO approximation and leverages our key theoretical insight that the optimal physics-informed distribution takes a closed-form Boltzmann form. Therefore, we employ calibrated Langevin dynamics with guaranteed convergence that balances distribution fidelity and physical law adherence without requiring architectural modifications or retraining. Our experimental framework features comprehensive four-par part evaluation strategy: First, we evaluate the effectiveness of physics knowledge incorporation within our framework, where the physics knowledge is known but the data is limited. Second, we assess multivariate time series forecasting capabilities on real-world applications featuring missing values and multi-resolution data. Later, we demonstrate PINFDIT's effectiveness as a generative model through comprehensive tasking, including synthetic generation, imputation, and anomaly detection. Fourth, showcase how our PINEDiT serves as a
model with the zero-shot setting.
The main contribution of our work can be summarized as the following threefold:
A Unified Framework for Imperfect Real-World Data: We propose PINFDIT to address the real-world scientific challenges of missing values, multi-resolution and multivariate data. With a systematically designed masking mechanism, PINFDiT jointly leverages the transformer's 's temporal modeling and diffusion's 's generative sampling capabilities in an end-to-end training pipeline, creating a robust framework specifically tailored for complex scientific time series analysis.
A Sorplliisttt--pppialii Inference Paradigm: We build a model-editing-ff physics knowledge injection framework under the lightweight Langevin-correction mechanism and the large-scale pre-trained model. This approach treats the model as a statistical generalist and applies a lightweight Langevin correction step to act domain specialist during inference. The implementation maintains theoretical convergence guarantees while ensuring generated time series adhere to fundamental physical laws without retraining the model.
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We validate PINFDiT through comprehensive experiments aligned with our foundational principles, evaluating its performance across physics-guidee guided forecasting, practical forecast- ing with miss value and multi-resolution sampling, synthetic generation, anomaly detection and imputation task. Moreover, we evaluate PINFDiT's ability in zero-shot settings. The results demonstrate state-of-the-arr performance, exceptional cross-task adaptability, and seamless integration of scientific domain knowledge, confirming the model's ' effectiveness for addressing complex temporal challenges in scientific applications.
RELATED WORK
Physsis-Informme Machine Learning. Physics has traditionally been injected into machine learning
systems during training by augmenting the loss with PDE residuals, in physics-informen neural net-
works (PINNs) (Raissi et al. 2019). operator learners such as DeepONet (Lu el. (02T) and FNO (Li
et al. 2021 more recent physics-aware transformers (Zhao et al. 2024). NeuralODEs (Chen et al. 2018 Beck et al. 2024) and their extension, NeuralCDEs Kidger et al. (2020b), explicitly model
latent dynamics ordinary controlled differential equations, thereby embedding continuous-time evolution into the network design. When the vector field is further regularised to respect conservation laws or Hamiltonian structure (Greydanus et al., 2019 Cranmer al., 2020), the resulting trajectories satisfy first-principle constraints throughout training, akin to PINNs and operator learners. In contrast, we follow the emerging line of inference-time debiasing that corrects a frozen surrogate without retraining: Simulation-Calibrated Calibrated SML (Fan et al.) 2025) and Monte-Care importance pinning (Lin al. 2024) exemplify this idea but have so far been limited owwdimensional surrogates. Diffusion- based generators have also been coupled to physical laws, e.g. SDEdit-PDE (Meng et al. 2022a) and FluidDiffusion (Qiu et al., 2024), yet these works focus on single snapshots or images rather than long, irregular time series. Orthogonally, simulation- -based inference (SBI) methods such as
SNPE (Lueckmann et al. 2017). CSBI (Gloeckler et al. 2024) and LFBC (Xiong et al. 2025) exploit
mechanistic simulators to learn parameter posteriors. Finally, hybrid forecasters that blend neural predictors with numerical weather models or other filters (Das et al.) 2024 Shi et al. 2023) illustrate the benefits of combining data-driven and physics- based cues. Distinet from all prior art, PINFDiT Langevin corrector:
marries a large-context transformer with a diffusion prior and a plug-and physics enters only through energy term, giving model-agnostic, zero-retraining route to enforce conservation laws while preserving the generative flexibility of foundation models.
Diffusion models for Time Series. Despite growing interest in diffusion models across various scenarios (Li et al., 2022a; Lu et al., [024] Sui et al.. [2024|]b) b), their application in time series analysis remains less explored compared pre-trained language models (Zhang et al., 2024 Gruver et al. (224:) Jin et al. [023] Pan et al., (2024] Ekambaram et al. 2024). Most existing studies also focus solely on forecasting and the choice backbone model also varies among VAE(Li et al.
2022b), R RNN(Rasul et al. (02T), and transformers. Recently, CSDI (Tashiro et al. 1021) first utilizes a diffusion model for time series imputation with a self self-supervised approach. SSSD (Alcaraz Strodthoff 2023) combines the structured state space model with the diffusion model for imputation.
ImDiffusion (Chen et al. (023) leverages diffusion models as time series imputers to achieve accurate anomaly detection. D'VAE Li et al. 2022b) proposes a generative time series forecasting method on top of VAE equipped with the diffusion model. Meanwhile, DiffusionTS (Yuan & Qiao 2024) incorporates decomposition into the diffusion model to improve interoperability. Although TSDiff (Kollovieh et al.) 2023) build a diffusion pipeline for multiple tasks with refinement, they still train different models for each task. Based on our knowledge, no unified diffusion transformer model has yet been explored for a comprehensive set of time series tasks. For thorough literature review on diffusion models in time series analysis, please refer to (Yang al. 2024).
METHODOLOGY
In this section, we begin by establishing a uniform problem setting for multiple downstream tasks, followed by comprehensive examination of our model architecture. We then explore the training pipeline with specialized mask strategies and observation conditions that enable effective self- supervised learning for time series data. Next, demonstrate how to incorporate physics information during inference to generate samples that better align with real-world constraints. Building on this foundation, introduce our plug-and-play physics injection component and provide theoretical
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(A)
Target Ouiput
(B)
Scale
Other Cond.
Embed
Scale
Senle, Shin
Projection Lager
Temporal Attention
Feed Forward
MLP
TimeDTT Flock
Diffusion
Step,
Embed
Scale, Shift
Scale, Shift
Eibeed
Pps
Layer Norm
Layer Norm
condition.
Emiheddnn Layer
Neised Target
Embeed
Cond Obs
Mask
Cond
Target
Observation pox
Target Mask
TimeDIT Model
( I. tideee
Time Series Mask Unit
Imputs with valuss, multiresslutimm..
Physics
Knowledge
Find Guual
Kix
E Fki0" [DDx(e()ll(())]
(C)
Figure 1: PINFDiT Architecture. (A): PINFDiT framework with diverse multivariate time series from different domains with multi-resolution or missing values; (B): Detailed structure of PINFDiT block; (C): Physies Injection: We employ physies knowledge a plugin mechanism, injecting known physical residuals F to refine predictions without requiring architectural modifications retraining.
adjustments showing its adaptability across various models. These extensions highlight the flexibility of our proposed approach, establishing it as a powerful model suitable for diverse time series.
3.1 TIME SERIES DIFFUSION TRANSFORMERS
Overview. Figure shows the overall framework of PINFDIT, Firstly, we establish Mobs and x based on inputs with varying shapes, missing values, and multi-resolution data. Then, the unified time series mask unit constructs M and adapts to diverse scenarios, generating x and x with shape I Bx I. where B is the batch size. Different from the channel- -indeeddndd patching token approach [Nc et al., 2023 Liu et al. 2024 Shi et al. 2024) vector quantization (Li et al. 2024), adopt a "What You See Is What You Get (WYSIW YG) design philosophy in our embedding layer, which directly maps tokens direct, contiguous arrays of x5 and the noised xha embedding vectors. After that, the output of the embedding layer fed into PINFDiT's diffusion pipeline, where xD integrated guide samples toward regions of high classifier likelihood, substantially improving output quality. Please refer to Section for the mathematical preliminaries of the conditional diffusion framework and the derivation the training objective.
Time Series Mask Unit. Our Time Series Mask Unit (TSMU) provides a powerful masking mechanism generating four specialized mask types: random mask M block mask M stride mask MS, and reconstruction mask Ree Leveraging diffusion models' inherent capability to progressively denoise targeted sequence regions with precise conditions, the TSMU enables selective focus on critical temporal patterns while maintaining contextual integrity. Furthermore, the unified masking approach eliminates the need for task-specific architectural modifications, enhancing the model's 's adaptability to diverse time series challenges.
Transformer- based Condition Injection. PINFDiT processes multivariate time series through transformer architecture that receives embedded noisy targets x!!"r and conditional observations x; Improving upon previous approaches (Peebles & Xie 2022 Lu et al., 2024), we directly inject the diffusion time step t into the target noise representation the entire series. For observation conditioning, rather than using simple concatenation (Rombach) al. 2022), implement adaptive layer normalization (AdaLN):
provides unit temporal information across
AdaLN(h, c) = LayerNoohh))) + Cahifts
where h represents the hidden state output from the previous layer. The parameters Cscale and Cahift derived from xi (Figure B)) and are essential for preserving temporal continuity and progression in the reconstructed series.
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Algorithm 1 Physics-Informedt Eieegyybbsedd Sampling
I: ær (0)
for =T...... do
3: ~~ *,(N0, ift 1, clsse 2 0)
4: 2t-1 var
Eo (0(2x)))) ozz
end for
for j == 0 1, ,.... ., do
7: m for
m1+1 = m + + K ( se + ar Vollog(a + 2200 0~ ~N00
fer
8: end for
9: return ZE
3.2 SEVSICS-INFORMED SAMPLING
Physics principles fundamentally govern the evolution of temporal signals in scientific phenomena like climate patterns and oceanographic data. In this section, we develop a decoding method ensuring that x"" generated by PINEDIT satisfies physical law constraints. Unlike the previous SBI solution (Xiong 2025 Linhart et al., 2024), which requires access to a faithful, differentiable simulator, we target the state-traje ctaeetrajy distribution directly by incorporating physics knowledge
ctt al.
energy- based prior during inference. This follows the paradigm of separating general model learning from targeted physical refinement at inference time, thus can be treated plugin mechanism requiring no architectural modifications or retraining.
Generalized Goal of Physics Injection for Time Series: We first start with brief introduction to physical laws and PDE. A generic form of physical law represented as a PDE that describes the evolution of continuous temporal signal x(u, r) over a spatial coordinate u and time is given by:
0x Ør
Ox
F(T,x, ou,' o,,uu,
2
Based on this PDE representation of physical knowledge, the consistency between the predicted time series x and the physics knowledge can be quantified using the following squared residual function:
K K(x lar, F) = -11 8x tar
Oxtar x2
F(T,
u, Su, ou,du, .I..
This function reaches its maximum when the predicted time series is perfectly consistent with the physical model, resulting in a residual 0. Using this metric K, physics knowledge can be integrated into a probabilistic time series foundation model p(x'"" x==") explicit regularization by solving the following optimization problem to obtain a refined model q(x!"r x=""):
g(x
max Bytting K (xxx F) aDxi("(x")) "x") (n")
(4)
where the first term represents the aforementioned physics knowledge metric, and the second term controls the divergence between (xx"Ixx"0 and p(x""" px")).
However, directly updating the model parameters optimize the above function is resource-consumingg To solve this issue, we derive the closed-form solution, which does not need to update the model parameters. The above optimization problem has closed-form solution as provided by the following theorem:
Closed-Form Langevin Correction via Boltzmann Energy Distribution:
Theorem 3.1 (Boltzmann Energy Distribution). The optimal ((x ) in Eq[4 is the Boltz- mann distribution defined on following energy function: E(xtor; xoo) K(xtar, F) (x"on) is:
a log fogg((x"")) p(xtar (x()), in other words, the optimal ((xtor(x<om)
1
q(x (xon))
(xp(K(xtar) F) a alg
(5)
where Z Teex(X(x (ar,F FF) log ((x )dx is the partition function.
The theorem illustrates that sampling from the Boltzmann distribution is analogous to incorporating physics knowledge into model edition. In the context of diffusion models, this distribution can be
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effectively sampled using Langevin dynamics (Stoltz et al. 2010):
X++ x)rrr + eV log ((x x + V2EO (0,
(6)
= =x + c VV lut, x + a (x]p("""">>>)) + V2c0, a ((
where o ~ (N(O,) 1). In diffusion model, precisely calculate the likelihood log p(x (x is in- tractable. To tackle this issue, following previous works (Kollovieh et al., 2023), approx- imate likelihood with the objective to edit the pre-trained diffusion model: log (( (xx00) =
(e..:ll/e((x" x - -l The approximation presented above constitutes the optimizable component of the evidence lower bound(ELBO). Algorithm I summarizes the comprehensive model editing process. By integrating physics knowledge through Langevin dynamics during inference, we achieve conssitetnn refinement without compromising the model's 's ability to learn from data and generate diverse. physically plausible scenarios.
We now present theoretical guarantees for our diffusion model's convergence properties and it can be easily extended other models:
Theorem 3.2 (Physics- Informee Inference Plugin Convergence). Let pe (x"") be the conditional distribution defined by a pre-trained diffusion model with score function co- Let F represent a physical law with residual function K ( F) in Eq. For the physics-informet plugin with step size and N refinement steps, the samples of Langevin dynamics Eq. 66 converge to the optimal goal of Eq. ( with convergence rale of:
DKL 14 10)=0(
+
Excore
where ON is the distribution after refinement steps and is the effective dimension of xar state space, Ebcore represents the squared error in the score estimation.
When we the step size (line 7 in Algo. 1) O(N-1/2) (Dalalyan & Karagulyan 2019), the converge rate can be ONN- The critical connection between statistical convergence and physical accuracy is established in Lemma [.3] which demonstrates that improvements KL divergence directly translate to enhanced physical consistency:
Lemma 3.3 (Residual-Variance Coupling). Let = 0,x F(-) be the physical residual of any sample x 9. If L-Lipschitz and the surrogate bias is 5, then for all q absolutely continuous || d') +.
w.r.l. s,, [ < 212 DKi(y
We establish a direct relationship between statistical convergence and physical consistency: each Vt-step improvement in KL divergence systematically reduces the variance of the physics residual, ensuring that our refinement process progressively enhances the physical validity of predic- tions. This relationship offers practitioners a clear interpretability pathway: improvements in model convergence directly translate enhanced physical consistency, allowing users to understand how refinement steps progressively reduce violations of physical laws. We extend the theorems to establish that PM (x" xx)0)) can represent the conditional distribution defined by other models M, making our approach model-agnostic. The complete model-agnostie proofs are provided in Appendix A
4 EXPERIMENTS
4.1
PHYSICS-GUIDE SCIENTIFIC TIME SERIES FORECASTING
We evaluate our model's ability incorporate physics knowledge into time series forecasting across six PDE simulators with multivariate ID systems modeling incompressible fluid dynamics. These systems are governed by the Advection, Burgers, Navier-Stokes, iiffusion-SSoppton, Vorticity, and Computational Fluid Dynamics equations, with the latter two reserved for further analysis. As shown Table [ PINEDIT consistently outperforms all competing approaches across nearly all metrics and equations. Our comprehensive comparison includes pre-trained foundation models (TimesPM, Moirai, Chronos), state-of-the-an linear/ransfommrrddffffusss bbssed deep learning architectures (DLinear, PatchTST, CSDI, DSffusionTS), physics informed models with embedded equations (NeuralODE, NeuralCDE), and inference-time time physics-guide approaches (SNPE. LFBC), included to benchmark against methods that similarly enforce physical consistency via simulators during inference. PINFDIT
physics-iiformed

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Published as a conference paper at ICLR 2026
Table 1: Comparison of model performance across different physical systems using both deterministic
metrics for accuracy and probabilistic metries (CRPS) for uncertainty quantification evaluation. Bold
indicates best result, Underline indicates the second best result. Same as below.
Model
Advection
Burgers MAE
Navier-Stokes
Diffusion-Sorptie
RMSE
MAE
CRPS
RMSE
CRFS
RMSE
MAE
CRPS
RMSE
MAE
CRPS
TTimesFM
0.0602
0,11428
0.0640
0.0278
0,0164
0.0822
0,0100
0,0076
0.0065
0.0407
0.0226
0.0279
Moirai-Small
0.1033
0.0738
0.1
0.1432
0.1193
0.5057
0.0368
0.0250
0.0243
0.0144
0.0096
0.0105
Moirai Base
0.0669
0,0481
0.0731
0.0972
0.0824
0.3701
0.0147
0.0107
0.0097
0.0044
0.0034
0.0037
Moirai-
0.0666
0.0481
0.0721
0.0365
0.0254
0.1227
0.0145
0.0097
0.0089
0.0052
0.0022
0.0025
Moirai MoE
0.0463
0.0300
0.0458
0.0275
0.0164
0.0827
0.0273
0.0199
0.0162
0.0023
0.0016
0.0019
Chromos-154 BB-B
0.0414
0.0258
0.0390
0.0202
0,0100
0,0474
0.0081
0,0055
0.0047
0.0019
0.0015
0.0018
Chronos-T3-S
0.0334
0.0218
0.0530
0.0424
0.0304
0.1366
0.0121
0.0080
0.0066
0.0028
0.0022
0.0023
Chromos-ool Bolt-T
0.0919
0.0691
0.1056
0.0776
0,0703
0.3232
0.0465
0.0366
0.0298
0.3995
0.1895
0.2235
Chromos-belt-B
0.1009
0,0706
0,1029
0.0205
0.0131
0.0634
0.0365
0,0285
0.0228
0.1375
0.0545
0.0792
Chronos-bolt-S
0.0678
0.0534
0.0776
0.0445
0.0351
0.1585
0.0378
0.0329
0.0261
0.1771
0.0717
0.0874
DLinear
0.3674
0.2943
0.4248
0.0287
0.0216
0.0975
0.9321
0.9101
0.6977
0.7898
0.7384
0.6754
PatcTTST
0.3767
0.3026
0.4410
0.0271
0,0208
0,0953
0.8972
0.8861
0.6928
0.76V8
0.7197
0.6549
CSDI
0.0118
0,01554
0.0176
0.0167
0.0076
0.0178
0.0094
0.0060
0 0.0055
0.0012
0.0008
0.0013
DiffusionTS
0.3393
0.2484
0.3540
0.0361
0.0263
0.1363
0.8267
0.6192
0.5063
0.0469
0.0358
0.0384
NeuralODE
0.3667
0.2968
0,4268
0.0284
0.0214
0.0966
0.9314
0.9045
0.6965
0.7860
0.7290
0.6707
NeuraLCDE
0.3690
0.3021
0.4314
0.0292
0.0221
0.1001
0.9318
0.9051
0.6972
0.7849
0.7327
0.6706
SNPE
0.0529
0,0422
0.0667
0.0472
0.0373
0.1903
0.1281
0.1134
0.0912
0.0557
0,0451
0.0482
LFBC
0,0352
0,0280
0,0512
0,0382
0,0294
0.1615
0.0739
0.0576
0.0491
0.0359
0,0287
0.0361
TSDiff
0.5128
0,5612
1.0000
0.1949
0.1670
0.9999
0.9212
0.9077
0.9989
0.7696
0,7373
1.0000
PINFDIT[W/ Phys)
0.0052
0.0040
0.0061
0.0136
0,00336
0.0163
0.0039
0.0031
0.0028
0.0057
0.0045
0.0050
PINFDIT
0.0039
0.0032
0,0050
0,0133
0.0035
0,0159
0.0037
0.0030
0.0027
0,0052
0.0041
0.0049
achieves remarkable improvements: compared to the best foundation model (Chronos-T5-BB, we reduce RMSE by 88.3% on Advection 35. 1% on Burgers, and 54.3% on Navier-Stokes Traditional deep learning models struggle with limited training data from the simulator, failing to learn meaningful representations for physies forecasting, as evidenced by their high error rates. Physics-informed neural networks that incorporate equations during training face similar data efficiency challenges. Among simulation-based inference methods, likelihood- LFBC outperforms SNPE by 33.5% on Advection, highlighting the difficulty of accurate likelihood estimation. PINFDiT, with its direct physics knowledge injection through closed-form Langevin dynamics, achieves superior performance without requiring retraining architecture modifications, demonstrating the effectiveness of our energy-based physics guidance approach.
In addition, we also added real-world experiment on climate forecasting. We use the ERA5 dataset for 2-meter temperature prediction and apply our PINFDIT framework, enforcing physical consistency via Navier-Stokes-basee constraints during inference.
Table 2: ERA5 climate forecasting (2-meter temperature), PINEDIT achieves higher ACC.
Model
6h
12h
18h
24h
NODE
0.82
0.68
0.69
0.79
ClimaX
0.92
0.90
0.88
0.89
Our experiments, evaluated using the A Anomaly Corre-
lation Coefficient (ACC) (Li et al., 2025) across mul- tiple lead times, demonstrate that PINEDiT outper- forms strong baselines and its own ablations, achiev- ing state-of-the-art accuracy on this challenging weather prediction task. These results highlight the effectiveness of our plug-and-plas physics-injectooe module for improving real-world climate forecasts, particularly under data scarcity and complex dynamics.
ClimODE
0.97
0.96
0.96
0.96
PINFDIT (wo Phys)
0,985
0.984
0.986
0.985
PINFDIT (Full)
0.987
0.987
0.987
0,987
4.2 PRACTICAL FORECASTING WITH MISSING DATA, IRREGULAR AND MULTI-RESOLUTION
To evaluate PINFDiT's performance in realistic scenarios, we conducted experiments incorporating three aforementioned challenges in real-world time series analysis across diverse domains from climate science to finance to healthcare: missing values (validated on Air Quality and MIMIC-III datasets), multi-resolution data (tested on NASDAQ), and irregularly sampled time series with varying time intervals between observations (cvaluated on three PhysioNet configurations). This comprehensive testing ensures PINFDIT can handle both technical challenges and domain-speciff nuances, validating its practical utility across diverse applications.
As shown in Table 3 PINFDIT consistently outperforms competing methods, achieving the best results in 19 out of 23 metrics across all datasets. For the Air Quality dataset, our model reduces MAE by 12.97% compared to the next best model (DiffTS), while simultaneously improving uncertainty
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Table 3: Forecasting results practical scenarios with both deterministic metrics (MAF/MSE) for
accuracy evaluation and probabilistic metrics (CRPS/CRPS sum) for uncertainty quantification.
Air Quality
MIMIC-III
NASDAQ
MAE/MSE
CRPS/CRPS_ sum
MAE/MSE
CRPS/CRPS sum _ssu
MAE/MSE
CRPS
DLinear
0,683/0.685
0.662/0.544
0.786/1.000
0.770/0.748
2.715/8.137
0.342
Neural ODE
0.678/0.679
0.657/0.529
0.784/0.999
0.769/0.733 733
3,227/11.155
0.426
Neural CDE
0.683/0.685
0.659/0.55 551
0.787/1 .002
0.771/0.754
3.319/11.816
0.439
PatchTST
0.685/0.683
0.664/0.564
0.778/0.987
0.771/0.721
3.182/10.635
0,410
GPT4TS
0.696/0.701
0.666/0.588
0.750/0.921
0.751/0.690
3.176/10.873
0.419
CSDI
0.539/0,554
0.598/0.620
0.551/0.681
0,504/0. 798
0.524/0,388
0.096
DiffTS
0.521/0.538
0.649/0.719
900
0.633/0.766
1,951/9.515
0.283
TimeMixer
0,691/0.697
0.667/0.576
0.769/0,981
0.776/0.724
3.267/11.511
0.432
TimeLLM
0.701/0.705
0.664/0.571
0.787/1.020
0.785/0.700 700
3.125/10.276
0,405
PINEDIT
0.457/0.354
0.554/0.522
0.517/0.534
0.599/0.649
0,516/0.418
0.091
PhysioNet(aa)
PhysioNet(b)
PhysioNet(e)
MAE/MSE
CRPS/CRPS_ sum
MAE/SSE
CRPS/CRPS
MAE/SSE
CRPS/CRPS
DLinear
0.686/0.758
0.764/0.812
0.733/0.922
0.794/0.799 793
0.715/0.813
0.767/0.797
Neural ODE
0.685/0.756
0.763/0.806
0.732/0.918
0.792/0.789
0.713/0.811
0.765/0.793
Neural CDE
0.688/0.754
0.763/0.799
0.733/0.921
0.792/0.786
0.713/0.814
0.765/0.791
PatchTST
780
0.769/0.812
0.733/0.932
0.791/0.77 775
0.714/0. 802
0.766/0.777
GPT4TS
0.697/0.772
0.767/0.809
0.734/0.921
0.795/0.796 798
0.713/0.817
0.770/0.768
CSDI
0.548/0.548
0.620/0.64 641
0.665/0.792
0.725/0.787
0.665/0.695
0.669/0.748
DifTS
0.610/0.742
1628/0.668
0,701/0.880
0.720/0.724
0.678/0.872
0.679/0.719
TimeMixer
0.692/0.775
0.763/0.805
0.734/0.920
0.794/0.798
0.707/0.800 805
0.757/0. 784
Timel
0.687/0.761
0.752/0.799
0.731/0.931
0.795/0.795
0.713/0. 800
0.757/0.754
PINEDIT
0,577/0.620
0.616/0.640
0.659/0.766
0.708/0.710
0.543/0.561
0.668/0.708
quantification with 7.3% lower CRPS. On MIMIC-III, PINFDiT achieves 6.17% reduction in MSE over the previous ofate-of-the-art (CSDI). The model's strong performance extends to financial time series, where delivers 1.53% improvement in MAE on the NASDAQ dataset. Most notably, on the challenging PhysioNet(c) configuration with irregular sampling, PINFDiT reduces MSE by 19.28% compared to the next best approach. These results demonstrate that PINFDIT not only achieves high accuracy in point forecasts but also provides well-calibrated probabilistic forecasts, effectively capturing the inherent uncertainties in complex time series data without requiring specialized designs for interpolation or imputation. This robust uncertainty quantification capability positions PINFDiT as a powerful tool for decision-making in uncertain environments across various domains.
4.3 GENERATIVE ABILITY ON SYNTHETIC DATA GENERATION AND IMPUTATION
Table 4: Imputation results averaged over four mask Table 5: Synthetic generation on 24-length ratios. Each cell is MSE/MAE.
multivariate time series.
Methods
ETT12
Waather
Flectricity
Metrie
Stocks
Air
Energy
SINFDIT_arrachh
13420,135
00400.130
0.01100036
1.869/0.13
0,1217
0.6969
GPT4TS
Wransformer
0.149/2.770
175
0.480.141
0L130A0.711
005588111
0.31..005
0.099/0.224
0.09002.07
DS
TAmeVAE
DiF-TS
0.0099
0.089
0.1987
0.1669
0.1227
0.2869
0.69%
0.2501
TimesNet
1.7892.187
0.499.11
0.030110144
0.0220.210
PINEDIT
0,0086
0.0087
0,1923
0,0053
PachTST
0.115/0.224
0.35/0.66
0.03611,555
1.0720..833
TimeGAN
0.2797
0.081
0.035
013305
Timer
0.14511223
0.077M..12
0.10801166
0.077/1.194
TimVVA
0.0069
0.2878
DAFTS
1.4000.134
0.0290.115
0.055A11199
0.075AB.165
DHE-TS
0.2262
0
0.2506
PINEDIT
0.036/0.122
0.01100111
0.01.00036
0.068/0.172
PINFDIT
0.1915
0.0455
0.0217
0,2489
Synthetic Generation Task: High-quality synthetic data generation is particularly crucial in scientific domains as it enables sewacy-preserving. research on sensitive datasets, facilitates model development in data-scarce fields. We conduct experiments to synthesize multivariate time series and evaluate performance using the discriminative score (DS) and predictive score (PS) metrics (Yuan Qiao 2024). Table demonstrates PhysDifIT's superior synthetic generation capabilities, with additional results using limited training data (5% and 10%) presented in Table 17 consistently producing more on challenging energy datasets. This demonstrates PINFDIT's strength in complex time series synthesis. Both qualitative and quantitative results confirm PINEDiT's superior ability to model intricate characteristics for realistic multidimennsioaal time series.
realistic samples than baselines
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Imputation Task: Imputation is essential in scientific research as it enables working with incomplete data due to sensor failures, sampling constraints, or measurement errors preserving valuable datasets that would otherwise be unusable and maintaining statistical power in experiments where data collection is costly or time-limited. We conduct experiments on six benchmark time-series datasets: ETTh1, ETTh2, Electricity, and Weather. We use random mask ratios {12.5%, 25%, 37.5%, 50%) following previous studies' settings (Wu et al., 2023). Table shows ssowss the imputation result averaged over the four mask ratios. PINFDiT demonstrates superior performance. Notably, PINFDiT achieved a 39% reduction in MSE and 22% reduction in MAE compared to the strongest baseline on the ETTh1 dataset. For full results on each mask ratio, please refer to Section C.4
For further anomaly detection task, please refer to Section C.3
4.4 GENERABILITY ON ZERO-SHOT SETTING AND ABLATION STUDY
Zero-shot capability is crucial in scientific domains where data col- lection is costly, time-consuming, or impossi- ble for new scenarios. Building upon previous success, we extended our investigation to zero- shot probabilistic forecasting, where PINFDiT demonstrates remarkable capabilities through our casy-to-implent pre-trained model design that employs masking strategies to accommo- date uniform inputs across diverse datasets. As evidenced in Table the PINFDiT achieves su- perior performance compared to models with fewer parameters equivalent zero-shot settings, outperforming state-of-the-art models on Solar, Electricity, and Taxi datasets with significant margins. These results establish PINFDiT as a significant advancement in time series modeling, particularly for applications requiring robust uncertainty quantification in novel scientific contexts.
Zero-shot forecasting:
Table 6: Forecasting results on CRPS sum LLgLL
notes for LagLLama; TMixer notes for TimeMixer;
TLLM notes for L.M.
Datasel
PINEDIT
Muirai (S/B/L)
Lag
TMiser
TLLM
Solar
0.424
0.884 / 0.948 1 142
0.000
0.999
0.997
Electricity
0.030
0.079 /0.092 10.009
0.065
0.302
0.303
Traflic
0275
0.403
Taxi
0.392
00428 00597
0.630
0.785
Exchange
0.019
0.007/ / /
0.024
0.079
0.076
Ablation Study: As shown in Table 7 As shown in Table 7 we conduct comprehensive ablation studies to validate the effectiveness of
Table 7: Ablation study on zero-shot forecasting
modelname's ' key components. We evaluate sev-
Dataset
PINEDIT
w/o Phys
wio RM.
w/ SM.
wlo BM.
eral variants: modelnamespace without physics injection (w/ Phys); without random mask- ing (w/o RM), stride masking (w/o SM), and block masking (w/o BM); alternative attention mechanisms including dual-attention (DA) and channel -wise attention (CW) compared to our temporal attention; different embedding approaches with patch tokens (PT) versus our direct embedding: and various observation conditioning methods via feature addition (Add) and cross attention (CA) versus our AdaLN approach. The full PINFDiT model achieves superior performance (MSE of 0.424 for Solar and 0.030 for Electricity datasets) compared to its variants. Among the masking strategies, stride masking proves to be the most crucial component, as its removal leads to substantial performance degradation (MSE increasing to 0.862 for Solar and 0.1 for Electricity). In the attention mechanism analysis, our proposed temporal-attentie architecture demonstrates consistent advantages over alternative designs, with patch token attention yielding the least favorable results (MSE of 0.874 and 0.145 for Solar and Electricity, respectively). Furthermore, the conditional AdaL design emerges as the optimal choice among various conditioning approaches, reinforcing our architectural decisions.
Solar
0.445
0.469
0.862
Electicity
0.030
0.053
0.035
0.037
0,101
Dataser
DA
CW
PT
Add
CA
Solar
0.467
0.461
0,874
0.677
0.711
Electrciity
0.037
0.039
0.145
0.079
0.077
CONCLUSION
In this paper, we introduce PINFDIT, a pioneering approach to creating a versatile and robust foundation model for various time series tasks under practical scenarios. By integrating transformer inductive bias with diffusion model, PINFDiT effectively captures temporal dependencies and addresses real world challenges unique to time series regarding multi-resolution and missing values as well as incorporating external knowledge. Our innovative masking strategies allow for a consistent training framework adaptable diverse tasks such as forecasting, imputation, anomaly detection, and

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Published as a conference paper at ICLR 2026
synthetic data generation, positioning PINEDIT bedrock of foundation model. We recognize some limitations of current work: first, we primarily explored common sequence lengths and did not assess PINFDiT's performance
very long sequences. While we have introduced randomness in prediction length and feature numbers up to a maximum, we aim to develop more scalable solutions for highly variable multivariate time series. second, there is high demand for deeply developing foundation models for multi-modal time series, allowing PINFDiT to utilize diverse data sources for enhanced performance.
ACKNOWLEDGEMENT
This work is partially supported by NSF Award #2425919, and NSF Award #2413417. The funding from these sources has been cornerttone in enabling us to bring our project to fruition. We are also deeply grateful to the anonymous reviewers for their rigorous review process. Their detailed comments and constructive suggestions have significantly contributed to the improvement of this paper.
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Appendix
Discussion Physics-Informed PINEDIT
19
Generalization on Proposed Physics Injection Method
19
Empirical Results on Convergence Characteristics
20
A.3
Robustness and Sensitivity Analysis
21
A.3.1
Sensitivity Langevin dynamics Refinement Steps.
21
A.3.2
Validity and Specificity of Physics Injection.
21
A.3.3
Robustness Physical Constraint Specificity on Climate dataset
22
A.4
Proof of Physics-Iniorma PINFDIT Theorem 3.1
22
Proof of Physics-Informed Refinement Plugin Theorem
3.2
23
A.6
Proof of Residual-Variance Coupling Lemma
3.3
26
Computation of Physics-Guided Gradients
27
A.8
PDE Systems in Synthetic Simulators
28
Real-world Climate and Weather Experiments
29
Background
30
B.
Preliminaries of Diffusion Models
30 ................................
B.
Problem Formulation: Conditional Time Series Generation
................................
30
B.3
Energy-Based Models and Langevin Dynamics
31
Detailed Experiment Results
31
Forecasting
31
Synthetic Generation
34
C.3
Anomaly Detection
35
Imputation
38
Analysis on PINFDIT
38
D.1
Ablation Study
38
D.2
Mask: Handling missing values
39
D.3
Condition scheme for PINFDIT
40
D.4
Noise Embedding Justification
40
D.5
Failure Scenarios Analysis
41
D.6
Dynamic on Model Size
Experiments setting
41
Baselines
41
Datasets
42
Metrics
43
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PINFDiT Paradigm on Training and Inference
45
Use of Large Language Models (LLMs)
47
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Figure 2: The two key parameters that govern CFD behavior: diffusion coefficient (1)) and oscillation frequency ((). Higher y) values produce smoother solutions with increased diffusion, while lower values create sharper gradients; higher values generate higher frequency oscillations with smaller structures, while lower values result in lower frequency oscillations with larger structures.
A DISCUSSION ON PHYSICS-INFORMED PINEDIT
The tension between physical constraints and learned distributions in PINFDiT is managed through sophisticated energy-based optimization framework that combines two key components:
the physics knowledge represented by function (xx";; F),, which measures PDE residuals for physical law conformity
the learned probabilistic distribution (200 from the diffusion model
This balance is achieved through an energy function:
Ela 3 K(a ((=""; F) log p(a'" x (o0))
where the parameter controls the off between physical consistency and distribution fidelity.
Rather than directly modifying model parameters, PINEDiT implements this balance through an iterative sampling procedure that:
starts with samples from the learned distribution
2. gradually refines them using physical gradients while maintaining probabilistic characteris- tics
This approach allows the model to generate samples that respect both the learned patterns in the data and the underlying physical laws without significantly compromising either aspect, ultimately resolving the tension through seooretically-groundde Boltzmann distribution as the optimal solution.
A.1 GENERALIZATION ON PROPOSED PHYSICS INJECTION METHOD
To validate PINEDiT ability to capture complex physical dynamics, we evaluated its performance zero-shot CFD prediction tasks across varying physical parameters. The demonstration of CFD)
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Table 8: PINFDITT' performance on sero-shot CFD with different parameters.
Model
MSE
RMSF
MAE
CRPS
CRPS_ sum
MSR
RMSE
MAF
CRPS
CRPS
LoW). <
0.00I(XLow)
1.0000Highh
0.001(MLow)
DDPM
0.0076
0.0874
0.0720
0.1012
0,0988
0.0035
0,0588
0.0485
0,0764
0.0622
Sid
0.0005
0.0009
0.0008
0.0002
0.0004
0.0001
0.0007
0.003
0,0002
0.0003
DDIM
0,0590
0.2428
0.2340
0.3663
0.3918
0.0370
0.1923
0.1843
0,3119
0,3320
DDIM Std
0.0014
0.0017
0.024
0,0001
0.0005
0.001
0.0017
0.009
0.0002
0.0004
TSDiff'
0.2844
0.5333
0.5256
1,0000
1,0000
0.2268
0,4762
0,4747
0.9999
0.9999
TSDifT_SI Std
6.0016
0.0017
0.007
0.0000
0.0000
0.002
0.002
0.01
0.0000
0.0000
PINFDiT
0.0046
0.0681
0.0558
0.0859
0.0755
0.0030
0.0547
0.0450
0.0714
0.0571
PINFDiT_Sid
0.0003
0.0004
0.0005
0.0001
0.0002
0.0003
0.0003
0.0005
0.0002
0.0001
0,00TO(
Low),
0.010O ligh)
0.0TOOK High), Ç
0.01000Tbgh)
DDPM
0.0052
0.0719
0.0598
T.09TO
0.0810
0.0042
0,0652
0,0530
0.0821
0.0719
DDPM_Sid
0.0006
0.0007
0.0011
0.0004
0.0003
0.0004
0.001
0.0035
0.0004
0.0003
DDIM
0.0464
0.2154
0.2043
0.3436
0.3694
0.0401
0.2002
0.1901
0.3227
0.3421
DDIM Std
0.0017
0.0021
0.016
0.0004
0.0002
0.0014
0.0019
0.0011
0,0002
0.0003
TSDift'
0.2497
0.4997
0.4858
0.9999
0,9999
0.2304
0,4800
0,4766
0,9999
0.9999
TSDifT Std
0.0018
0.002
0.004
0.0000
0.0000
0.0012
0.0015
0.013
0.0001
0.0001
PINEDIT
0.0046
0.0679
0.0559
0.0857
0.0764
0.0038
0.0613
0.0493
0.0772
0.0661
PINFDiT_SH
0.0004
0.0004
0.0003
0.0002
0.0001
0.0002
0.0004
0.001
0.0002
0.0002
NS MSE
10 ID-WorcclyyWe WANE
10-
6.00975
6.9054-
9.2
600850
30/00 1o
8.0023
6.0003
0.5-
02-
600875
0.000-
6.004-
0/0825
3.33
0.23
6.33
0.28
0.25
0.30
3.23
925
0:35
0.28
3.50
Figure 3: Empirical Convergence Analysis of NS and Vo-viciiy Models. Both results demonstrate linear relationships consistent with theoretical expectations, with the NS simulation showing stronger linearity but higher absolute error values compared to the ID- ID-Vorticity simulation, reflecting differ- ences in their underlying physical complexity.
flow can be found at Figure We systematically tested two key parameters that govern fluid behavior: diffusion coefficient (17) and oscillation frequency (C). Higher 1) values produce smoother solutions with increased diffusion, while lower values create sharper gradients; higher values generate higher frequency oscillations with smaller structures. while lower values result in lower frequency oscillations with larger structures. As shown in Table 8 PINFDiT consistently outperformed baseline models with different sampling strategies (DDPM, DDIM, and TSDiff) across all parameter configurations, demonstrating its robust ability adapt to different physical regimes without retraining. Notably, PINFDiT maintained superior performance even in the most challenging scenario with high oscillation frequency (< = 0.0100) and low diffusion (17 = 0.0010), where sharp gradients and complex small- scale structures coexist. These results confirm PINFDiT'S 's effectiveness in incorporating physical constraints and capturing the underlying dynamics of complex fluid systems, highlighting its potential for scientific applications requiring accurate simulation of physical phenomena.
A.2 EMPIRICAL RESULTS ON CONVERGENCE CHARACTERISTICS
The convergence plots in Figure 3 comparing the Navier-Stokes (NS) and ID-Vorticity simulations demonstrate strong alignment with the theoretical 1/2) convergence rate expected for numerical simulations. Both models exhibit a clear linear relationship when error metrics (MAE and MSE) are plotted against -1/2 as indicated by the red dashed trend lines. Notably, the NS simulation displays nearly perfect linear scaling for both MAE and MSE, suggesting the numerical errors decrease predictably as the number of time steps increases. In contrast, the ID- Vorticity model shows slight deviation from perfect linear convergence, with flattening of the curve at intermediate values of N. This behavior indicates that the ID- Vorticity simulation may be approaching a lower error bound more rapidly than the NS model encountering additional error sources that aren't 't.
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purely time-step dependent. While both methods converge the theoretical ((N-1/2) rate, the ID- DD-ootticity simulation achieves lower absolute error with the same computational budget. These results confirm that both models follow the expected theoretical convergence rate, while revealing important differences their numerical behavior that reflect the underlying physical complexity and implementation characteristics of each approach.
A.3 ROBUSTNESS AND SENSITIVITY ANALYSIS
A.3.1 SENSITIVITY TO LANGEVIN DYNAMICS REFINEMENT STEPS.
We further investigate the robustness of our physics-informer refinement module (Langevin dynamics) with respect to the number of refinement steps K. Table Tableereports reports the RMSE across three datasets for varying K. We observe that the performance consistently improves as K increases from 10 to 20, reaching an optimal range. Crucially, the model remains stable for K >> 20 without significant degradation divergence. This demonstrates that our inference- time optimization is robust and does not require hyper-sensitive sensitive tuning of the step count to yield performance gains.
Table 9: Robustness of Physics-Infore Informed Refinement. Comparison of RMSE performance across different numbers of Langevin refinement steps (K). The method achieves consistent improvements around 20-30 steps and maintains stability beyond that point.
Dataset
K = 10
K = 20
K = 30
K = 40
Advection
0.0039+000002 0.0047++0.0002
0.0043 0.043
0.0045
Burgers
0.0143+0.0003
0.0136+10.000
0.0133=0.0000
0.0135-10.00%
Navier-Stokes
0,0041+0.0003
0.0037=000001
0.0038=0.0001
0.0039:10.0002
A.3.2 VALIDITY AND SPECIFICITY OF PHYSICS INJECTION.
To verify that physics-injection module provides legitimate domain-specifts guidance rather than merely acting as generic smoothing filter, we conducted two sensitivity analyses. These experiments test the model's behavior when the injected physical laws either mismatched to the data or applied to non-physical domains.
Impact of Mismatched Physical Laws. We analyzed the performance on the Advection dataset when injecting the correct physical law (Advection Equation) versus an incorrect law (Burgers' Equation). As shown in Table TO utilizing the correct PDE significantly reduces the error (25% improvement). Conversely, injecting the incorrect Burgers' dynamics actively harms performance, increasing the RMSE by 17% compared to the baseline. This confirms that the performance gains are derived from the correct physical inductive bias and that the module is sensitive to the validity the governing equation.
Table 10: Robustness to Physical Mismatch. Comparison of performance on Advection data when injecting correct vs. incorrect physical laws. Metrics reported are RMSE (L).
Data Source
Physies Configuration
RMSE
Rel. Change
No Physics (Baseline)
0.0052+10.002 0.0039-10.0002
Advection
Correct (Advection PDE)
+25.0%
Incorrect (Burgers PDE)
0.0061+0.001
-17.3%
Application to Non-Physical Domains. To investigate whether the Langevin correction acts as a spurious "smoothing" operator, we applied fluid dynamics constraints (Advection and Burgers) to the NASDAQ financial dataset, which possesses no underlying fluid physical basis. As presented in Table II the application of physics constraints yields benefit and results in slight degradation of Mean Absolute Error (MAE). This suggests that our method does not produce spurious positive transfer; it functions effectively only when the data supports the underlying physical hypothesis.
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Table 11: Sensitivity on Non-Physical Data. Performance on financial data (NASDAQ) when forcing fluid dynamics constraints. Metrics reported are MAE (D).
Data Source
Physies Configuration
MAE
Rel. Change
No Physics (Baseline)
0.516++0.002
NASDAQ
Advection PDE
0.526:t0.004
-1.9%
Burgers PDE
0.537=00.005
-4.0%
Generalization to Autoregressive Foundation Models. To validate the model-agnostic capability of our framework, we integrated the physics-informed refinement module into Chronos-TS-Sman leading autoregressive foundation model. We approximated the score function from the predicted quantiles and applied Langevin dynamics (Eq. 6 as a post-processing step on the Advection dataset. As reported in Table 12] the physics injection yields consistent improvements across all metrics (CRPS, MAE, RMSE). The inclusion of standard deviations (computed over 5 random seeds) confirms that these gains are statistically significant and robust, distinct from random sampling noise.
Table 12: Model-Agnostic Refinement, Performance comparison on the Advection dataset using the Chronos-TS: Tr-onos-T--Smlll backbone. Results are reported as Mean + Standard Deviation.
Model
Method
CRPS (1)
MAE (L)
RMSE (L)
Baseline
0.0330 0.0002
0.0218 + 0.003
0.0334 + 0.0008
Chronos- T5-S
+ Physscss (Ours)
0.0313 + 0.0001
0.0197 + 0.002
0.0322 + 0.0005
A.3.3 ROBUSTNESS TO PHYSICAL CONSTRAINT SPECIFICITY CLIMATE DATASET
To validate that our framework actively enforces specific physical dynamics rather than acting generic regularization, we conducted counter-factual experiment the ERA5 dataset. We compared the standard PINEDiT (using the correct Temperature Coveecion-Diffusiin equation) against variant enforcing the incorrect Navier-Stokes Momentum conservation the temperature field. As shown in Table 13 applying the wrong physical law significantly degrades performance (ACC drops from s=0985 0,.954), confirming that the model is sensitive the physical validity of the constraint. Conversely, the correct physics injection yields consistent improvements over the base model, demonstrating that our method successfully steers the generation toward physically consistent atmospheric states.
Table 13: Physics Specificity Analysis. Anomaly Correlation Coefficient (ACC) on ERA5 2- meter temperature forecasting. We compare the base model against injection of the correct physics (Advection) and incorrect physics (Navier-Stokes Momentum). Results are reported as Mean + Std Dev.
Model Variation
6h
12h
18h
24h
PINFDiT (w/o Phys)
0,985 0.002
0.984
0.003
0.986
0.002
0.985
0.002
PINFDiT (NS Momentum)
0.954 + 0.004
0.956 = 0.005
0.955 - 0,004
0.954 + 0.003
PINFDIT (Advection)
0.987 0.003
0.987
0.002
0.987
0.001
0.987 0.001
A.4 PROOF OF PHYSICS-INFORMED PINEDIT THEOREM 3.1
Theorem A.1. The optimal g(x!") in Eq 3 is the Boltzmann distribution defined on the following
energy function:
E(x xx))) = (x F) + a p(x (x
(8)
in other words, the optimal g(xtar (x)m) is:
1
q(x tar xaalx = Z exp( exp(K(x'"rr (xr F) + log loxpppp'''" (x
(9)
where 7 Texp(K(x (xttar F) log po (xcon) ) ddxtar is the partition function.
/exp(
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Proof. Let consider the objective function:
O(q(y(z)) Byra(ula) K (1) a aDKx(a(i)))lll(l)e)
By K(y) a /olll)) log ((ya))
dy
p(y/z)
(10)
[ alggp(y)))) log
We try find the optimal ((ylz)) through Lagrange multipliers. The constraint of the above objective function is that g x) is valid J, a(v x) = 1. Thus, the Lagrangian is:
L(g(vlz), = ((vl))) ((ylz)(K + logp(elz) log logg(v/=))dy 2 d(vlx)dy 1)
(11)
+ a
log (()l)()))))))))))))))))))))){< - Ag(y/=)]dy +
We define f(g(yla), y, A) be the density function of any fixed distribution defined the support set of y. Therefore, L(g(v(z), A) = J f(g(v(x), y, A)dy. According to Euler- Guler-Lagrange equation, when the above Lagrangian achieve extreme point, we have:
+ logp(ylz)) a log (()lx)) Nk + Ah(2)). where h(y)
(ιy) + a logp(ylx) logg()/z) X 0f 0q
(12)
Thus, have:
log ((v)2)) K(y) + (ogpp(l))) log (()lx)))))))))))))))()))
<vvlz) exp( 1
K(y) + logp(ulx) A - 1)
(13)
1
exp( 1) exp(
exp( - k(y) + log p(yl)))
Meanwhile, since I, d)yy 1, we have:
/
expp (xpp_K(<) K(w) logp(y/zx A 1)dy
y
(14)
1
eepp
(exp(=K()) K(y) log
exp( 2 + 1)
a
Thus, we have exp( A + 1) = I, exp( ( (v) - logp(y|z))dg = Z, leading to:
Z exp (y)(K()) +alg = / exp(((v)) + a llg
(<01=)
(15)
Note that F represents fixed knowledge for a given domain so it remains constant during the optimization process and doesn't need to considered as variable the proof.
A.5 PROOF OF PLYYSICS-INFORMEL REFINEMENT PLUGIN THEOREM 3.2
We now present a theoretical framework for applying chysics-informe refinement to any time series prediction model, establishing its model-agnostic nature.
Assumption (Regularity Conditions). We assume that K(x; F) and log are L-smooth (have Lipschitz gradients) and satisfy mild dissipativity/corcvvii condition such as (x, (x())) > m|x |x| - for U -K - a log РM, where m > and are constants.
Assumption 2 (Score Approximation). For any model M, we assume the score function can be approximated such that V/V log PM V log PM 1a S wwor-
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Discussion of Regularity Constants (m, b, L). In theoretical analysis, we carefully distinguish between three key constants in the convergence of Langevin Monte Carlo (LMC) and plug-and-play refinement for sampling:
L. (Smoothness/I .ipschitz Linschit Gradient): The Lipschitz constant of the gradient (x),,, i.e., [V(x) (y))l S T.I[x yll for all x. This bounds the maximum curvature and determines the sensitivity of the potential's gradient, and is required for standard LMC convergence proofs.
(Strong Convexity/Coerrivitt):: The strong convexity parameter, used in the dissipativity condition (x, (x)) 2 m||x|] - b. For globally strongly convex potentials (b 0), this controls the contraction rate and determines how well-behaved the energy landscape is at large scales.
b (Dissipativity Offset): The offset > quantifies the size of regions in parameter space where the potential may deviate from strong convexity (e.g., have flat or mildly non-convex regions), generalizing the notion of strong convexity admit more complex but still well-controlled landscapes.
We stress, following best practices Dalalyan & Karagulyan (2019); Erdogdu et al. (2022); Zhang et al. (2023), that the convergence rate of LMC is governed by the condition number L, m, not by the difference (m - L) or any related expression. This clarifies that for quadratic/strongly convex potentials (as in many PDE-constrained or physically regularized cases), the convergence is exponentially fast with rate proportional to m, and larger L (less smooth or stiffer systems) will slow convergence.
For only dissipative potentials (b > 0), global exponential convergence cannot always be guaranteed. but the system remains ergodic and convergence rates are still quantifiable, with 6 determining how much the potential can deviate from ideal convexity. This theoretical framework allows PINFDiT and related physics-informed informed diffusion models rigorously justify refinement across wide range of practical multivariate time series and PDE scenarios. Now we continue our convergence proof.
Theorem A.2 (Physics-Informed Inference Plugin Convergence). Let Pas (xtar|xeon) be the conditional distribution defined by any time series model M (diffusion model, transformer, RNN, etc.). Let F represent a physical law with residual function K K(x!"r; - 0x2" F(I, x u, Du," Su,0u, >l
"y 82
Under Assumptions 1 Hand 2 for the physics-informed plugin with step size e 1(N-1/2 and refinement steps:
3+1 =X"" + +XVK(x);; acV log PM (x] x²"") + / or
(16)
where oi -N(0,)) I), the resulting samples converge to the distribution q" (xtrr( that minimizes:
C(g) - [ (x'"r; F)] + aD (gDLlg(x""rrae (xtum)) || pM (xmm)))
(17)
with convergence rate of:
deff + -
DRL (1 < O
(18)
VN
where qN is the distribution after N refinement steps and daff is the effective dimension of the state space after considering any dimensionality reduction techniques employed.
Proof. The proof follows standard Langevin dynamics convergence results (Dalalyan & Karagulyan 2019), adapted to our setting:
Target distribution: The Boltzmann distribution representing the optimal balance between physics compliance and model fidelity, where Z is the normalization constant:
q" 10r X con)
F) + Pxr(px (x
(19)
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Langevin dynamics convergence: For the Langevin dynamics:
xj + VVlogq (x "x=") V
(20)
is known from statistical physies that these dynamics sample from the distribution q" in the limit of infinite steps. Specifically, the dynamics are simulating the stochastic differential equation (SDE), where W is standard Wiener process:
dx; = V log q" (x "dddt + v dW,
(21)
Convergence rate: Under the regularity conditions in Assumption and score approximation in Assumption for Langevin Monte Carlo (LMC) with step size e, the convergence rate is:
denc N
DKN 11 s So
+ denc² E
(22)
Optimal rate: With e = G(N 1/2) Dalalyan Karagulyan 2019)
deff
(aI |l 11) <
= O(N
(23)
Scoore
ignoring dimension-dependent factors for simplicity. The additional error term is bounded for model M and does not affect the ((N-1/2) convergence rate
Model-dependent approximation: For different model types, approximating V log PM (x!ar x will vary:
Lemma A.3 (Score Approximation for Different Model Architectures). For the following model architectures, the score function log Pas (x"" "x"" can be approximated as follows, each with bounded error Escore under appropriate conditions:
1. Difffusion Models: For diffusion model with noise prediction network ED.
v log PM (xx
Eo(xe, t, x
(24)
of
with ervor bound Escore = O(lec e" l12), where e" the optimal score function.
2. Autoregressive Models: For an autoregressive model with token probabilities
p(z (e x
V Pag (x|x ~ V., log logp(r,/ x x"))
(25)
with error bound Escore ( when using finite differences for approximation.
3. Energy-Based Models: For an energy-based model with energy function Eg(x, xeon).
V log Pas (x]x = (.n)
(26)
with no approximation error when the energy function is differentiable.
4. Variational Autoencoders: For VAE with encoder qs (2|x) and decoder po (x(22).
V llgg (x (x|x cm") x Bengg(=lxx.. V log po xx|2,x"00 x0")
(27)
with error bound Escore O(D O(DrI 20n |ppp |=]]](((xxx))))))
Remark 1 (Non-Con -on--Convexity). When K or log PM is non-convex, Langevin dynamics can still converge with the stated rate, but constants in the bounds may depend exponentially on the energy barriers in the landscape, potentially affecting mixing times in practice.
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Remark 2 (Model-Agnostic Nature). The convergence guarantees in Theorem 3.2 depend only properties of the physics operator F, the number of refinement steps N, and the score approximation enror Eware- They independent of the specific architecture, training procedure, internal structure of the model M. This establishes that our physics-informed plugin approach is truly model-agnostic.
Unlike previous approaches that incorporate physics constraints directly into specific model archi- tectures (e.g., physics-informed neural networks physics-constrained transformers), method provides a model-agnostic refinement plugin that can be applied to any time series model that provides way to approximate its score function. This fundamental shift in approach allows practitioners to of-thheart ahe advances in time series modeling while still ensuring physical consistency, without requiring specialized architecture modifications
leverage state-of-the,
A.6 PROOF OF RESIDUAL-VARIANEE COUPLING LEMMA
Lemma 3.3 connects KL convergence to physical accuracy, showing that improvement in KL directly improves physics compliance. This addresses the key concern: "Does better convergence actually mean better physics?"
Lemma A.4 (Residual- Variince Coupling). Let djx F(-) be the physical residual of any sample x ~ q. If F is T-Llipschitz and the surrogate bias is 5, then for all q absolutely continuous w.r.l. G,
[arg[]] S
2L DKL(a l1 () +4L26
(28)
Implication: every --sstep improvement in KL directly squeezes the variance of the physics residual.
Proof. By definition, the variance of the residual under distribution is:
Var, = E.y || -
[[]1]]
Let F 0,x F(-) be the residual under the target distribution q². By the definition of q², we know that E.. [|[**11 v 82, where represents the bias of the surrogate model.
Now decompose the expected squared residual:
/
3 |+]
(29)
/ 17(())12 (xx g(xx dx
(30)
(x())
Egg [1F(X)]2 g(x)
(31)
9-' (x))
Using Pinsker's inequality, we have:
fa(x) (x)) dx (2Dx(())))
Since F' is L-Lipschitz, we know that for any and y:
[F((x) Fly) S L||x - yll
Combining these results with the triangle inequality, we can bound the difference between the expected squared residuals:
< 2T.2 10222at fuxx) {x)) dx (32)
F [ - E.g [.
< 21. 22Dx((g))))
(33)
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Therefore:
Var r x [
(34)
< Eyp ||F + +22 (alra.(
(35)
s 8 + 212 2 (allo))
(36)
Using the inequality vI S 1 x/2 2/2 for x 2 0, we get:
2 (Kxi,(ull
Var, r] s 2T, 1
(37)
2
8 + 2L -2L2 -2L2DKL (glld))
(38)
8 + 2 (al(ol
(39)
where in the last step we used the fact that for small enough 8, we have 8² + - 4L
Thus, we have established:
Var
< 2L DRI(q |1 ()) +4L28²
This shows that as the KL divergence between g and q" decreases at rate 1/2),/2), the variance of the physical residual also decreases at the same rate, directly linking statistical convergence to physical accuracy.
A.7 COMPUTATION OF MRYYICS-GUIDDE GRADIENTS
core component of our inference-time adaptation is the computation the gradient V where represents the physical energy function defined by the squared PDE residual. As noted in Section 3.2. our method does not require a differentiable simulator (which would require solving the PDE forward); rather, it requires only differentiable residual function.
Given a discrete time series sample æ E RT generated by the reverse process (where T and C denote time steps and spatial/feature dimensions, respectively), compute the gradient as follows:
1. Finite Difference Discretization: Since x: a discrete tensor, we approximate the continu- the du ous partial differential operators (e.g., 'BF Brr 2h" using standard finite difference schemes (FDM). These operations are implemented as fixed convolution kernels acting on æ.
For the temporal derivative, we typically use a first-order backward difference:
Ex
a:
ærii
(40)
At
For spatial derivatives (if applicable the dataset structure), central difference schemes, e.g.:
8 x 0s2
æij + 1] 2x [1]
1]
(41)
(As)²
2. Residual Computation: We substitute these discrete approximations into the algebraic form of the governing PDE (Fiq. 5) to compute the residual field R(x). The energy is defined the norm of this residual, E(x) || a:) l1². Importantly, because the finite difference operations are linear transformations (convolutions) and the PDE algebraic combination is composed of differentiable operations (addition, multiplication), the resulting scalar (xx) is fully differentiable computational graph with respect to the input æ.
3. Automatic Differentiation:
Finally, to obtain the guidance term VaE(x) required for the Langevin dynamics correction, we utilize standard automatic differentiation (autodiff)) frameworks (e.g., torch. aut ogradd.. This allows us backpropagate the error through the finite difference kernels directly to the input sample x, enabling efficient physics-basd based guidance without the need for training surrogate model forward PDE solver.
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A.8 PDE SYSTEMS SYNTHETIC SIMULATORS
Distinction from Conventional Numerical Solvers. It is important to clarify why we compare PINFDiT against deep learning baselines rather than traditional numerical PDE solvers (e.g., Finite Difference or Finite Element Methods). Conventional solvers are designed primarily for forward problems where initial and boundary conditions are perfectly defined. They are not naturally equipped to handle the inverse-like challenges of real-world scientific data, which often contain missing values, sparse observations, or irregularities.
Therefore, the appropriate baselines for our work are physics informed learning methods, such as NeuralODE (Chen et al. 2018), Neural CDE (Kidger et al. (0200), and ClimODE (Verma al.) 2024). These methods share our goal of learning dynamics from observational data while respecting physical constraints. However, PINEDiT distinguishes itself from these baselines in two critical ways:
Robustness to Imperfect Data: While ODE-based methods often struggle with discrete, noisy, non-uniform data, diffusion-based framework inherently handles missing values and irregular sampling as a conditional generation task.
Inference- Time Injection: Unlike NeuralODEs, which require physics to be embedded dur- ing the expensive training phase, our method utilizes inference-tine injection via Langevin dynamics. This plug-and-play" capability allows us to enforce or swap physical constraints without retraining the model.
We use the following equations to generate samples with 40 spatial resolutions and 192 timesteps for evaluating and also generate the training samples for deep learning methods.
The Burgers Equation is:
Du
8u
+ u
U
= 0
(42)
Ot
Oz
Or.2
where is the diffusion term. We set the (diffusion term) as 0.1 and randomly sample a combination of sine waves as initial status
The Advection equation:
Ou
Ou
(43)
Ot Oz
where e is the advection speed. We set c = 1.0 and used randomly placed Gaussian peaks as initial conditions, generating challenging translational dynamics with preserved signal shapes.
The Diffusion-Sorption equation can be expressed as:
du
ju
kgu
(44)
dz
where u is the concentration, D is the diffusion coefficient, k2 is the sorption rate coefficient. Initial conditions are set as a Gaussian distribution:
u(x, 0) 0.5)2
(45)
The boundary conditions are zero-flux (Neumann boundary conditions):
Oul
Ou Fx
r 0
(46)
dr
where L is the domain length.
The Kolmogrov Flow is a specific case of Navier-Stokes (NS) equation. More specifically, is described by:
uar, (x,y,22t) = (-
(- Or abb Dy Ox 0)
(47)
where the psi is the flow function. It is usually set
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(v,,, = Asi((kx) cos(zy + wl)
(48)
where A, k w are hyperparameters.
The Vorticity equation is:
Ow + (u VJw 2
(49)
01
where w represents vorticity, u the velocity field, and v the kinematic viscosity coefficient. This equation describes the evolution of vorticity in fluid flow, capturing the rotational motion central to turbulence formation.
The Computational Fluid Dynamics (CFD) equation implemented in the code can be expressed as:
au
Du
8²u Oz
=sin({x)
(50)
Ōt
dr
where:
y is the viscosity parameter (analogous in the Burgers equation). F(x) sin((x) the forcing term derived from Kolmogorov flow. Initial conditions are set as:
w. 0) = sin(x) + 0.5 sin(()) 2x) + <(x)
(51)
where e(z) random noise sampled from normal distribution / (0, 0.25).
A.9 REAL-WORLD CLIMATE AND WEATHER EXPERIMENTS
For real-world climate forecasting, we evaluate PINFDiT on the ERA5 reanalysis dataset using the WeatherBench2 (Rasp et al. 2023) framework. The experiment focuses on 2-meter temperature (t 2m) prediction at a 5. 5.625 spatial resolution (corresponding to 64 x 32 grid points). The data is split according standard protocol (Li al. 2025). training on the years 2006-2015, validation on 2016, and testing on 2017-2018.
Physical Constraints. To ensure physically meaningful predictions for the 2-meter temperature (42m) forecasting task, we incorporate a physics-guided inference module based on the advection- diffusion (continuity) equation. This formulation aligns with the physical principles used in state-of- the-art atmospheric modeling works such as ClimODE (Verma et al. (024). The governing equation applied is:
8T = - VT) T(V 0t
(52)
where 7' represents the temperature field and v represents the wind velocity vector, derived from the dataset's u-component (u10) and v-component (v10) of wind 10 meters. This equation explicitly captures two critical atmospheric dynamics:
Transport Term -(v VT): Represents the spatial movement of temperature carried by the wind flow (advection).
Compression Term -V(V v): Accounts for temperature concentration dispersion caused by the convergence or divergence of the air mass.
Balancing Transport and Diabatic Processes: important to note that near-surface temperature is not a strictly conserved quantity due to diabatic source/sink terms (e.g. solar radiation and surface heat flux). PINFDiT addresses this by treating Equation 52 as soft regularization constraint via the energy function E(z) K(x; F') + a log (zr).. The learned diffusion prior log po(x) is responsible for capturing complex diabatic processes (such as diurnal heating cycles) from the data, while the physics residual K (a: F) penalizes deviations from consistent atmospheric transport. This synergy allows the model to respect fluid dynamics without forcing unphysical conservation in the presence of external heating.
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Metrics and Fair Evaluation: Evaluation is conducted using the standard- ized Anomaly Correlation Coefficient (ACC) (Li et al. 2025). implemented via weatherbenchi metrics ACC (elimatology),
All models, including baselines such as ClimODE, are assessed over identical test intervals and spatial resolutions by leveraging WeatherBench22s robust evaluation pipelines to ensure fairness and compatibility across methods.
B BACKGROUND
B. PRELIMINARIES OF DIFFUSION MODELS
In recent years, diffusion models have emerged promising approach to generative modeling. A diffusion process is a Markov chain that incrementally adds Gaussian noise to data over a sequence of steps, effectively destroying the data structure in the forward process and reconstructing the data structure during the reverse process.
The forward process adds noise the data Xo a series of timesteps i according variance schedule Br, resulting in a set of noisy intermediate variables X1, x2, Xy. Each subsequent x is derived from the previous step by applying Gaussian noise:
q(x,
x)) = (x /
(53)
The reverse process aims to denoise the noisy variables step by step, sampling each x,-1 from the learned distribution po (xx-11 xt). This distribution, modeled by neural network parameterized by 0, approximates the Gaussian distribution:
(x xx) = (x (40 (x., f), (x,, t))
(54)
By iterating this reverse process from = T down to t = 0, the model gradually reconstructs the original data from noise. Learning clean xT through the reversed diffusion process reduces to building a surrogate approximator to parameterize (x, t) for all t. The reverse process learns to predict the mean and covariance of each intermediate distribution, effectively approximating the original data distribution.
B.2 PROBLEM FORMULATION: CONDITIONAL TIME SERIES GENERATION
In the context of this work, we focus on conditional generation, where the goal is to approximate the distribution of target time series segment given context segment (observations).
We denote a multivariate time series as X = { E R where K is the number of features and L is the length of the time series. Each individual entry æi, represents the j-th feature at time step i, for i E (1,..., I.} and j E {1,.... K}. We define an observation where mi,j = 0 if æij missing, otherwise, ma, = Let xghs o denote the observed subsequence; x denote the target subsequence of x which could be forecast target or imputation target or the whole sequence depending on the task. Let xi denote the unmasked partial observations x9 which acts like self-conditions for the masked area x Let us use all subscripts of x to denote diffusion timestamp, and a subscript of O means no noise has been applied the original data. Formally, the goal of this research is to approximate the true time series distribution given the conditional information qx (xf x with model distribution px(x (x0)).
Problem Definition:
mask Mobs {m,,) E [0,1]
Conditional Reverse Process. We extend the diffusion framework to this conditional setting. The forward process is applied only the target tar while the condition con remains clean and provides guidance at every step of the reverse process:
po OxT
p (x²
Po tar
xtar xcan)
(55)
pe (x
X
x ron) == N (x pe (x
x con)
Eg
(x]
,t
.on)
Here, x ~ (0, I). The denoising network b4 takes the noisy target x!"r, the diffusion index t, and the clean context x inputs predict the clean mean.
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Training Objective. We train the denoising model μsl by optimizing weighted mean squared error loss. This loss is derived from the variational lower bound (ELBO) and focuses on matching the mean of the posterior distribution:
I8 oxe [lle(xx"" x0 (x( ; x6") )|?]
(56)
where pxx"",, x represents the tractable mean of the true posterior distribution ((x By minimizing this objective, the model learns to reconstruct the target trajectory consistent with the observed context.
xD x
B.3 ENERGY-BASED MODELS AND LANGEVIN DYNAMICS
Our tmysics-inforree' inference framework builds upon established principles in Energy-Bain Beeegy-Based Models (EBMs) and Product of Experts (PoE). Here, we briefly review the foundational concepts that enable our model-editing-fees approach.
Product of Experts and Energy Composition. The core of our inference strategy relies crarbuttonss-scccffccllll specifically, a learned data prior and a physical constraint. This is formally
combining distributions- grounded in the Product of Experts (PoE) framework introduced by [Hnnon| 1999), which states that high-dimensional distributions can be modeled as the product of several simpler "expert" distributions. In the context of EBMs. this compositionality allows for the combination of independent energy functions Du & Mordatch 2201)). Given two independent energy functions Edata (I) and Ephysios (2). the optimal joint distribution follows a Boltzmann distribution defined by the sum of their energies:
p(r)) exp (- Fiooaa))))) exp (- (Faataa Eephysies (x)))
(57)
Recent works have leveraged this property for implicit generation and compositional modeling Du & Mordatch| 2019); Grathwohl et al. (202T), demonstrating that complex constraints can be enforced at inference time without retraining the base model.
Langevin Monte Carlo Convergence. To sample from these composed energy distributions, we employ Langevin Monte Carlo (LMC). The convergence properties of LMC for log-coneave and non-log-concave distributions are sell-established, Dalalyan & Karagulyan (019) provided rigorous non-asymptotic guarantees for LMC convergence under Hessian Lipschitz conditions. Subsequent research has extended these bounds to broader divergence metrics, including Chi-squared and Rényi divergences Erddogu al. (2022), and improved discretization analysis for underdamped dynamics Zhang et al. (2023). Our theoretical analysis in Section 3.2 adapts these standard convergence results to the specific context diffusion transformers constrained by partial differential equations (PDEs), ensuring that our iterative refinement reliably converges to the physically consistent distribution.
C DETAILED EXPERIMENT RESULTS
C.1 FORECASTING
We comprehensively evaluate PINFDiT in zero-shot forecasting environments against leading ar- chitectures, encompassing both zero-shot and full-shot models. Using popular benchmark datasets for long-term deterministic forecasting that exclude source overlap with pre-training data, PINFDiT demonstrates exceptional performance across varying prediction horizons (96-720 time steps) as shown in Table [4] and Table [5] The model achieves superior Mean Squared Error (MSE) and Mean Absolute Error (MAF) metrics compared all baselines, particularly excelling on the ETTh dataset where it consistently outperforms all comparative models.
Building upon this success, we extended our investigation to probabilistic forecasting, where PINFDiT demonstrates remarkable capabilities. As evidenced in Table [6] zero-shot PINFDiT achieves superior performance compared to models with fewer parameters in equivalent zero-shot settings. We further fine-tuned PINFDiT, motivated by both its strong zero-shot performance and the inherent challenges in probabilistic modeling, where models typically struggle with calibration issues-eine either producing overconfident predictions with overly narrow distributions underconfident predictions
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Trndon
Tree/d
122
(a) PCA sine dataset
10% T5
TrxeAA
TreeAC
100
600
(b) PCA on stock dataset
(c) PCA on air quality dataset
211
015
3.5
2.04
12:0
1075 (8.
H420 800 6T0
1ED
+037 220
120
1008
(d) PCA on energy dataset
Figure 4: PCA evaluation of synthetic time-series data (TSD) generated by PINEDiT versus baseline methods on four benchmark datasets.
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Table 14: Full results of long sequence forecasting experiments evaluated diverse datasets, where bold indicates the best performance and underlined indicates the second best performance. Bascline results are sourced from Woo et al. (2024a).
Zemsehhu
Pull-shut
PINEDIT
MoIRAlm
MoIRAlLap
iTausforue
TamesNet
PatchTST
Crosdfrmmrr
Diiimar
SCINst
FEDleenne
96
0.325
0.575
0,354
0,350
0.385
0.384
0.414
0,423
0.479
0.385
0.554
0.37%
193
IL347
0.500
0.425
0.440
0.441
0.446
0.460
0.471
0.525
0.437
0.710
014:3)
(MSE)
336
0.414
0.347
0.412
0.413
0.400
0.456
0.705
0.51
0.487
0.331
0.891
0.501
0.500
0.631
0.570
0.565
0.519
0.481
0.78
0.366
0.499
0.30m
Akg
0.356
0310
0141
0.257
0.281
0,222
0.287
0.340
0.302
0.745
0,400
0,707
0.358
ETT12
(MSR)
336
720
192
0.316
0.341
0.447
0.00
0.380
0.540
0.340
0,311
0.304
0.347
0.377
0.404
0.425
0.580
0.427
0.402
0.462
0.452
0.383
0.436
0.31
0.573
1.003
1.104
0.74
0.538
0.643
0.31
0,511
0.477
0.560
1,240
1840
0:499
0.4%
0.463
A1g
10340
0.355
0151
1051
0.402
0.445
0,310
0.419
0.429
0.485
0.474
01631
0.468
(MAT)
336
720
0.395
11:10
0.429
0.611
0.450
0.473
0.434
0.508
0.458
01:51
0.469
01:500
0.466
0.438
0.621
0.546
0.515
01554
0,515
0.659
0.639
0,009
0.465
0,500
Asg.
0.403
0U424
0.438
0.469
0.448
0.450
0.455
0.522
0.507
0.452
0.647
0.46
0.594
0.323
0.374
0.400
0.459
(MAZ))
720
0.41%
0.44]
0.313
0.416
0.401
0.426
0.421
0,393
0.445
0.657
0.431
0.466
0.511
0.763
0.670
0.637
0.638
0.744
0.467
0.444
Axg
01303
0.379
0.382
0.376
0.407
0.897
0.407
0.684
0.50
0315
0.73
0.409
Table 15: Performance comparison of recent models ETTh and ETTh2 datasets
Diinet Mane
Horans
11MF-300
TIME 300
TIEEXDE.
Tos.IM
Trec2M
Mires
Traa use
192
0.347
3.184
0.341
3.366
0.65
0.405
0.68
0.533
0.822
0.503
0.500
0.001
Du544
733
13.33
0.443
8.433
3.27
0.381
0.311
0383
0.852
0.500
02266
D4077
Agg
0.330
0.40
3412
01273
0207
0.211
0444
1123
17501
172
0.310
0.31
0364
2347
0.31
0.50
0354
0.776
0.564
0.236
03003
0300
0348
730
01447
0435
0331
2435
0331
0143
0.50
0.500
0.51
0.00
0425
0.90
D4te
13.50
033
12233
1251
0330
0350
0.300
0.04
0412
0.451
0.54
0266
0426
0446
6410
B447
731
01220
3.471
5.555
0462
0.502
0.51
0.523
0.91
0.58
0344
068
96
01331
0350
0351
0353
0337
0.59
0396
0.300
030
036
0361
0300
0366
MAB
0.115
0.455
4335
0,63
0.21
5400
0.681
0.81
0.0
D414
0441
0.511
2.455
8.50
6-51
8134
0.53
0.17
6511
6:50
0466
with excessively wide intervals due to insufficient domain-specific knowledge. The CRPS sum evaluation reveals that fine-tuned PINFDiT surpasses specialized probabílistic forecasting models across multiple datasets, demonstrating exceptional performance in both zero-shot generalization and fine-tuned specialization. These results establish PINFDIT as a significant advancement time series modeling, particularly for applications requiring robust uncertainty quantification.
Forecasting Setting. For the physics-guided forecasting, we generate the evaluation samples via the simulator as described in Section For the practical forecasting, the dataset information can be found at Table 2n] and the setting can be found at Section E.2 To enable fair CRPS comparisons, we followed the standard approach used by Moirai Woo et (2024a) and other time series foundation models: we augmented deterministic baselines with lightweight Student-t distribution heads that predict parametric distributions with the point prediction as the location parameter. These distribution parameters were learned jointly by optimizing negative log-likelihood during training, and CRPS was computed using the closed-form expression for continuous distributions.
Table 16: Forecasting results on CRPS sum for both zero-shot and full-shot settings
Setting
Datasel
PINEDAT (2S)
TEMPO
Moirabsi
Metrait))
Maisuit.)
LudiLaMA
TimeNliser
TimeL1M
Timee
Solur
0.424
0.561
0:045
1.04
0,690
0.907
Electricity
0.830
0.072
0.065
0.302
0.303
0.301
Zero
Traflic
0.351
0,391
0.275
0.400
0.368
0.384
Tast
0,392
0.40
04661
0.42%
0.597
0,600
01285
0.242
0.788
Eschange
0.019
0.50
0,007
0.12
0011
0.004
0.079
0.076
0.073
Setting
Dataset
PINFINT (PS)
PatcTST
Latent OUR
GPTTT
TraneAA
TimeGrad
CSDI
Difthdon
0.286
Electricity
Full
0.095
0.056
0.64
0.000
0.097
Tusi
0.123
0.181
0.179
0123
Exchange
0.005
0.013
0.005
0.007
0,009
Zero-shot Forecasting Setting, For the zero-shot forecasting task we utilized five widely-used open datasets evaluate probabilistic time series forecasting performance. These datasets were
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Table 17: Limited observation data Synthetic Generation results 24-length multivariate time series. Discriminative and predictive scores are calculated as described in (Yon et al. 2019).
0.05
0.1
Metric
Methods
Sine
Air Quality
Energy
Sine
Air Quality
Energy
TimeGAN
0.120(0.043)
0.500(0.003)
0.50000. .00)
0.067(0.028)
0.492(0.003)
0.500(0.000)
Diseriminative Score
TimeVAE
224
0.498(0.001)
0.50000.000)
0.495(0.002) (.499(1.0012)
0.499(0.001)
Diffusion-TS TS
0.037(0.013)
0.496(0.003))
0.498(0.0055
(.001(0
0.494(0.001)
0.494(0.011)
PINEDIT
0.031(0.007)
0.456(0.0033
0.472(0.000)
0.030(0.009)
0.437(0.004)
0.447(0.002)
TimeGAN
0.231(0.007)
0,148/0.029 029)
0.30800 006)
0.200(0.002)
0.130(0.029 (.00)
0.302(0.004)
Predictive Score
VAE
0.328(0.008) (.251().003)
0.308(0.014) 0.238(0.002) 0.296(0.001)
0.288(0.001)
Diflusion-TS
0.196().003)
0.111(0.004)
0.333(0.018) 8)
0.188(0.001)
0. 1.102(0.010
0.340(0.01))
PINEDIT
0.194(0.001)
0.089(0.005)
0.335(0.008)
0.192(0.000)
0.070(0.007)
0.318(0.005)
collected in GluonTS (Alexandrov et al. 2020) and have been previously employed in (Tashiro ct al. 2021 Salinas et al.] (219)). The task for these datasets is to predict the future L2 steps given the observed L1 steps. We set L1 and L2 values based on previous studies (Tashiro et al. 2021 Salinas et al., 2019). For training, we randomly selected L1 + L2 consecutive time steps as a single time series and designated the last L2 steps as forecasting targets. We adhered to the train/test splits used previous studies and utilized the last five samples of the training data as validation data. For the full-shot setting, trained separate models different datasets. Due to the large number of features in multivariate time series, we adopted subset sampling of features for training. For each input, split them into subsets based on their order. If the last subset was smaller than the fixed shape, we applied padding to ensure equal input sizes across all subsets.
C.2 SYNTHETIC GENERATION
We use 80% of all data for training and evaluation of the same data. For the air quality dataset, previous methods did not carefully use the -00 values as a placeholder for missing values. In our experiment, we masked all the -200 values for PINFDIT and baselines that support masks. For baselines that do not support mask, we replace -200 with the mean value. minmax scaler is used for all models. Diffusion-TS TS uses a different normalization scheme between -1 and We replace its normalization scheme to be minmax scaler to ensure fair comparison. Figure 4 shows the PCA plots for all datasets and baselines. The visual comparison also validates the superooity of PINEDiT.
We also run the generation experiments with the limited data fine-tuning in Table 17 The generation experiments with limited data fine-tuning demonstrate PINFDiT's s superior performance across various datasets and evaluation metries. Comparing TimeGAN, VAE. Diffusion-TS. and PINFDIT on sine, air, and energy datasets with 5% and 10% training data, PINEDiT consistently achieves the lowest Discriminative Scores, indicating its ability generate the most realistic time series. In terms of Predictive Scores, PINFDiT outperforms or matches other models, particularly excelling in the air dataset. Notably, PINFDIT's performance remains robust or improves when increasing from 5% to 10% training data, showcasing its effectiveness in data-scarce scenarios. These results highlight PINFDIT's capability to capture complex temporal patterns and generate high-quality time series data, even with limited training samples, making it promising tool for various time series generation tasks.
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C.3 ANOMALY DETECTION
We conduct experiments on five real- world datasets from industrial appli- cations: MSL, SMAP, SWaT. SMD, and PSM. While PINFDIT's s founda- tion capabilities benefit most tasks, anomaly detection's 's unique require- ment to model dataset-specific nor- mality led us employ from-scratch
Table 18: Threshold Sensitivity Analysis on Anomaly Detec- tion Performance evaluated Fl score
Threshold 99.5
99
98
97
96
95
MSL
83,9
89.33
90.1
88.17 17
85.28
82.84
PSM
96.32
97.57
96.78
95.72
94.66
93.61
training, highlighting the importance of matching deployment strategy to task characteristics. We introduced
SMAP
97.08
95.91
93.23
90.33
87.64
85.09
SMD
83.28
82.07
76.61
70.73
65.71
61.24
SWAT
97.6
96.46
93.49
90.74
88.0
85.42
spectral residue (SR) transformation at the preprocessing stage of PINEDIT. This transformation helps to conceal points most likely to anomalies and their immediate neighbors. The number of neighbors affected is controlled by the hyperparameter nneighhor- The SR method utilizes Fourier Transformation to convert the original time series into a saliency map, thereby amplifying abnormal points, as detailed in (Ren set the sequence length to be 100 identify anomalies using the 99th percentile of reconstruction errors. During evaluations, we apply standard anomaly adjustments suggested by (Xu et al.] 2018). As demonstrated in Table C.3 PINFDiT outperforms baseline models on four of the five datasets. In particular, PINFDiT 23.03 points of improvement in terms of F1 score on the SMAP dataset compared to the previous best baseline. In addition, PINFDIT consistently outperforms both TimeMixer and iTransformer across all datasets, with particularly notable improvements on SMAP (95.91 vs 67. 077663666.77 76) and SWAT (97.57 vs 88. (8.84/92.63). These comprehensive comparisons against the latest models demon- strate PINFDiT's effectiveness as a unified framework for time series analysis, often achieving state-of-the-art performance while maintaining broader applicability across diverse tasks.
et al. 2019; Zhao et al. 2020). Consistent with prior methodologies,
Anomaly Detection Threshold Our comprehensive analysis of threshold selection in Table T8 revealed that higher percentile thresholds, particularly the 99th and 99.5th percentiles, consistently yield superior performance. While we observed a systematic degradation in detection accuracy as threshold values decrease, maintained the 99th percentile threshold to ensure fair comparison with existing methodologies. This decision reflects our commitment to methodological rigor, as optimizing threshold values based on test set performance would introduce bias in the comparative analysis. Our approach prioritizes consistent experimental conditions across all evaluated methods, enabling meaningful benchmark comparisons while acknowledging the impact of threshold selection on detection performance.
Table 19: Anomaly Detection result on 100-length multivariate time series. We calculate Precision, Recall, and Fl score as % for each dataset. '." notation in model name stands for transformer. Bold indicates best result, Underline indicates the second best result. We replace the joint criterion in Anomaly Transformer with reconstruction error for fair comparison.
Methods
MSL
SMAP R
SWaT
SMD
PSM R
1st PI Count
Metrics
R
F1
P
F1
F1
F1
F1
PINFDIT
89.33
93.35
97.57
GPT(6)
82.00
82.91
82.45
90.60
60.95
72.88
92.
96.34
94.23
$8.89
$4.98
86.89
62
95.68
97.13
TimesNer
89.54
75.36
$1.84
90. 14
56.40
69.39
90.75
95. 65.
93.02
X7.91
81.54
84.61
51
96.20
97.
PatchTST
88.34
70.96
78.70
90.64
55.46
68.82
91.10
80.50
85.72
87.26
82.11 14
84.62
84
93.47
96.08
ETStomer
85.13
84.93
85.03
92.
55.75
90.02
80.36
84.91
87.44
79.23
83.13
99.31
85.28
91.76
FEDlomer
77.14
80.07
78.57
90.47
58.10
70.76
90.17
96.42
95.19
87.95
82.39
85.08
97.15
97.23
LighTTS
82.40
75.78
78.95
92 58
55.27
69.21
91.98
94.72
93.33
87.10
78.42
82.53
98.37
95.97
97.15
incar
7X
Autofommer
27.27
80.92
79.06
58.62
71.12
89.85
95.
88.06
82.35
85.11
99.08
88.15
93.29
Anomaly.
79.61
87.37
83.31
91. 85
58.11
71.18
72.51
97.32
83.10
88.91
82.23
85.49 49
68. 35
94.72
79.40
TimeMixer
89.72
75.42
81.95
89.51
$4.
67.63
91. 56
86.
88.84
86.00
71 50
78.33
99 18
87.74
93.11
Thrnsformer
86.16
62.64
72.54
90.69
52.82
92.21
93.06
92.63
86.92
77.75
98
95.32
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value
40
(a) Imputation results ETThl dataset.
value
-1.25
time
(b) Imputation results ETTh2 dataset.
value
-1.5
-2.0
-2.5
-3.0
20
40
60
80
Imputation results ETTml dataset.
20
40
60
80
time
(d) Imputation results on ETTm2 dataset.
Figure Visualization of imputation task on ETT datasets. This figure illustrates PINFDIT's performance, with red x's marking observed values, blue dots showing ground truth points for interpolation, a green line representing PINFDiT's mean of interpolation, and green shading indicating its estimated uncertainty intervals.
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value
60
80
time
(a) Imputation results electricity dataset
-0.70
valie
-0.75
0.80
0.85
time
(b) Imputation results weather dataset
Figure 6: Visualization of imputation task on electricity and weather datasets. This figure illustrates PINFDiT's performance, with red x's marking observed values, blue dots showing ground truth points for interpolation, green line representing PINFDiT's mean of interpolation, and green shading indicating its estimated uncertainty intervals.
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Table 20: Full result of imputation task.
Teac
Caraao
Teaodho
Ligres
BLace
Scnach
Mat Kadi
M4l
MA2
N55 364t
MAI
HBL MAL
MAS
500. MAG
NSE MA1
MA2
NSL.
MA1
HSz MAR
M42
HSE
MAR
MAD
INSE
MAE
275
1432
8.%%
9332
H16 033
8:5
0.39
230
151 830
1000 6100
0122
1115
8:34
3.00
506
R465
0:53
1178
8.30
028
6214
821
41 828
620
840
9464
0257 630
033 628
8.6
8215
0235
0.79
8.28
0.30
1160
256
R404
680
3334
3168
0263
0.57
014
0150
637
836
0137 624
8220
6168
044
0336
8:31
8.34
3130
(141 8:0
133
0122
1001
1.100
816
100
61N
Red 0372
1335
1155 836
1085
0.30
6117
8:11
1140
8136
HA4
6.177
6130
6334
1063
0220
1002
0.10
6:68
1003
1.31
1.00
6316
1eg
14221
0:00
211
010
0100
010
1100
0
0144
6.5
011
1006
0.76
0351
1100
1m0
0100
0.01
190
12.55
1004
1115
07.16
114
211e
110
010
000
0:0
3:00
1:00
Arg
0100
8:30
1.244
1.50
12.5% 1040 8:30
2134
0.4
0.00
0214
1003
6:38
1148
17.4 HAxt
0:1
8:13
1.714
1100
0.10
0.504
8.00
1222
0220
0230
1:30
1174
12-5
2144
015411
1000
0000
200 HUCH
6154
1137
0151
832
1007
87h 1122
K7a
8461
18
2114
0100
600
011
0000
0:30
01-
C.4 IMPUTATION
For visual representation of PINFDIT's imputation capabilities, we have plotted the results in Figure and Figure 6 which clearly illustrates the model's s accuracy in reconstructing missing data points across different datasets and missing data ratios. The imputation task results, presented Table C.3 demonstrate PINFDiT's superior performance across various datasets and missing data ratios. All baseline models are trained a full-shot setting, while PINFDiT leverages a pre-trained foundation model, fine-tuning it on realistic datasets. PINFDiT consistently achieves the lowest Mean Squared Error (MSE) and Mean Absolute Error (MAE) scores in most scenarios, outperforming state-of- the-art models such as GPT2, TimesNet, and PatchTST. Notably, PINFDiT's performance remains robust even as the proportion of missing data increases from 12.5% to 50%, showcasing its ability to handle substantial data gaps effectively. The model's imputation accuracy is particularly impressive for the ETThI, ETTh2, ETTml, and ETTm2 datasets, where it maintains a significant lead over other methods. PINFDiT demonstrates superior performance on most datasets, achieving significant improvements over Timer, TimeMixer, and iTransformer, particularly on ETT datasets where we see reductions in MSE by up to 60%. PINEDiT maintains strong overall performance while offering greater versatility
D ANALYSIS ON PINFDIT
We present a comprehensive analysis of PINFDIT's design space, conducting systematic comparisons across different architectural variants. To ensure fair evaluation, all experiments maintain consistent training configurations, utilizing the same checkpoint and number of training steps. This rigorous experimental setup allows us isolate and assess the impact of individual architectural components while controlling for training conditions.
Table 21: Ablation study on zero-shot forecasting
Dataset
PINEDIT
w/o Random Mask
w/o Stride Mask
w/o Block Mask
w/o Phys
Solar
0.424
0.465
0.469
0.862
0.445
Electricity
0.030
0.035
0.037
0,101
0.033
Dataset
Dual-attention
Chanoei-wise
Patch Token
Additive
Cross-attention
Solar
0.467
0.461
0.874
0.677
0.711
Electricity
0.037
0.039
0.145
0.079
0.077
D. ABLATION STUDY
Our comprehensive ablation studies, detailed in Sections E1, E2, and E3, systematically evaluate PINFDIT's s architectural choices. In Section E1, with particular emphasis on the Transformer design strategy, explore PINFDIT's temporal-wise attention mechanism and compare it against alternative
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approaches, including channel-wise attention and dual attention mechanisms (as discussed in Yu ct al., 2024)). The analysis demonstrates that temporal-wise processing significantly outperforms traditional patch-based tokenization approaches, achieving substantially lower enror rates.
This performance disparity can be attributed to two key factors:
First, while channel relationships exhibit model-specific variations, temporal patterns provide more universal characteristics across time series data, enabling better generalization. Second, patch- based approaches introduce additional hyperparameter dependencies (patch length and stride settings) that compromise the model's universal applicability. These findings validate our design choice of temporal-wise processing as a more robust and generalizable approach for time series modeling. The empirical results strongly support our architectural decisions, demonstrating that PINFDiT's temporal-focuse design effectively captures universal temporal dynamies while maintaining model flexibility across diverse applications and domains. In addition, the Informe- component yields consistent performance improvements across all datasets, with notable enhancements in Electricity and Solar predictions, underscoring the value of incorporating physical constraints during inference.
D.2 MASK: HANDLING MISSING VALUES
Time Series Mask Unit. The Time Series Mask Unit is a key component of our model, designed enhance its versatility and performance across various time series tasks. This unified mechanism incorporates multiple mask types that seamlessly integrate with the model throughout its lifecycle from self-supervised task-agnostic pre-training to task-specific fine-tuning and inference. The time series mask unit generates four distinct mask types: random mask M block mask M stride mask MS, and reconstruction mask M During task-agnostic pre- pre-training, these masks help the model develop robust and generalizable features from the input data, improving overall time series representation. In task-specific training, the masks adapt to the unique requirements of common downstream tasks such as forecasting and imputation, enabling the model to specialize effectively.
Given x E Rgxx xL, the random mask R can be generated by:
M (c. ()_ Zz, > r, K ,2 ~ Uniform(0,)
(58)
otherwise,
where r is the mask ratio. For task-specific training and inference, we allow the user to supply customized imputation masks, which replace the random position masks, that could handle the naturally missing data and multi-resolution cases. In addition, block mask M can be generated via:
{1
j<L-l otherwise,
M (x,
(59)
where / is the predicted length. This mask offers flexibility across different stages of model devel- opment and application: during pre-training, random exposes the model to various forecasting horizons, while in fine-tuning and inference, fixed I aligns with specific task requirements. More- over, stride mask MS, a variant of M is designed for intermittent placement within time series during aask-agnostic pretraining:
( 0
[ mod 2 0
M$ (z. Tolockk))
(60)
otherwise,
where Botock is the number of blocks; b sequence. improves the modeling of temporal and cnter-correlated dependencies by integrating information across non-contiguous parts of time series, leveraging neighboring values as additional context. addition, reconstruction mask M generation and anomaly detection. It allows direct generation of synthetic data or calculation of anomaly scores for each temporal position by comparing original and reconstructed series.
is the length of each block; is the index of the
employed for tasks such as synthetic data
Our experimental design leverages naturally occurring missing values inherent in real-world datasets, primarily arising from irregular sampling rates and multi-resolution data collection processes. This approach authentically validates model robustness against genuine missing data patterns rather than artificially generated scenarios. PINEDiT incorporates a comprehensive masking strategy that aligns with three well-established missing data mechanisms: Missing Completely Random (MCAR)
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using uniform distribution- baaebd random masks, Missing at Random (MAR) employing block and stride masks to capture structured patterns and dependencies between non-contiguous observations, and Missing Not at Random (MNAR) utilizing reconstruction masks with -infses-iffomee sampling for scenarios where missing patterns correlate with unobserved variables. These mechanisms are simultaneously applied through self-supervised learning, enabling robust representation learning without requiring explicit knowledge of the underlying missing data processes. Our comprehensive ablation studies in Table 21 demonstrate the criticality of each masking strategy, where removing any mask type leads to performance degradation, with future masks showing the most significant impact. These findings validate our integrated approach to handling diverse missing data scenarios in time-series analysis.
D.3 CONDITION SCHEME FOR PINEDIT
As mentioned in Section 3.1 AdaLN's superior performance stems from its ability to dynamically adjust feature distributions across different layers while maintaining computational efficiency. This approach aligns well with the inherent nature of time series data, where temporal dependencies typically exhibit gradual rather than dramatic changes both seen and unseen time steps. We conducted comparative experiments to evaluate different conditioning mechanisms in PINFDiT:
Additive conditioning, which adds conditional information directly to the diffusion input;
Cross-attention, which uses conditional time series as keys/values and noisy time series as queries to fuse conditional information;
Token concatenation, which concatenates conditional time series with noisy time series at the input level before PINEDiT processing.
The experimental results (Table 21 across Solarand Electricity datasets consistently show that AdaLN achieves superior performance compared to the next best alternative. This significant performance gap validates our choice of AdaLN PINFDiT's primary conditioning mechanism.
D.4 NOISE EMBEDDING JUSTIFICATION
PINFDIT's noise embedding ap- proach plays multiple key roles the diffusion modeling framework. The diffusion process operates directly in continuous embedding space, allow- ing for smoother transitions between noise levels and better preserving the inherent time dependence, thus en- abling the model to learn more bust representation of the underlying time series structure. This approach has several technical advantages (Ho al. 2020 Peebles & Xie 2022 Lu et al., 2024): the embedding space provides a continuous representation in which the diffusion process can op-
Table 22: Performance metrics for weather and ecl datasets on different model sizes.
L
MSE
MAB
MSE
MAE
MSE
MAE
0.125
0.029
0.033
0.029
0.026
0.025
0,024
0.250
0.031
0.033
0.033
0.029
0.028
0,027
Weather
0.375
0.034
0.037
0.036
0.033
0.031
0,031
0.500
0.031
0.041
0.042
0.039
0.036
0,036
Avg
0.031
0.036
0.035
0.032
0.030
0.029
0.125
0.051
0.148
0.050
0.144
0.048
0.140
0.250
0.061
0.163
0.060
0.158
0.154
0.375
0.074
0.181
0.071
0.175
0.069
0.170
0.500
0.090
0.202
0.087
0.197
0.190
Avg
0.069
0.174
0.067
0.169
0.065
0.163
erate more efficiently. The direct em- bedding of noisy samples helps prevent the embedding space from collapsing during training. From practical point of view, this approach allows for parallel processing of multiple time steps, handles varying degrees of noise through unified framework, and makes the diffusion process more stable compared to traditional generation methods. In addition, the embedded noise representation allows for the seamless incorporation of physical constraints and maintains temporal continuity while pro- gressively denoising. thus contributing to better quantification of the uncertainty in the generated samples.
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D.5 FAILURE SCENARIOS ANALYSIS
PINFDIT's s performance shows notable degradation in three key scenarios: highly irregular sampling rates deviating from training distributions, complex non-stationary patterns underrepresented in pretraining data, and domain-specific patterns requiring expert knowledge beyond general time series characteristics. As shown by SMD dataset for anomaly detection (Table C.3) where it achieves 83.28% Fl score versus GPT2's 86.89%. This dataset represents cloud server machine metrics with high-frequency sampling and complex feature interdependencies. Additionally, when dealing with extremely short-term patterns or highly localized anomalies, specialized architectures like GPT2 that focus intensively recent temporal context may outperform PINFDiT's more holistic approach, as our diffusion-based generation process may occasionally smooth over abrupt local changes. These limitations, primarily stemming from the model's dependence on learned foundational patterns, become particularly relevant in specialized industrial applications and unique financial scenarios. Understanding these boundaries crucial for informed model deployment decisions and highlights promising directions for future research.
D.6 DYNAMIC ON MODEL SIZE
The experimental results demonstrate a clear correlation between PINFDIT's model size and its imputation performance across different datasets and missing data ratios. As shown in Table 22 as the model size increases from Small (S) to Big (B) to Large (L), we observe consistent improvements in both averaged Mean Squared Error (MSE) and averaged Mean Absolute Error (MAE) metrics. The Large model consistently outperforms the Small and Big variants across all scenarios, with the most significant gains observed in the weather dataset. Notably, larger models (B and L.) show better resilience to increased proportions of missing data compared to the Small model. The improvement more pronounced for the weather dataset than for the ecl dataset, suggesting that the benefits of increased model size may vary depending on the nature and complexity of the time series data. The consistent performance gains from B to models indicate that PINFDiT's architecture scales well with increased model size. These findings suggest that increasing PINFDIT's 's model size is an effective strategy for improving imputation accuracy, particularly for complex datasets or scenarios with higher proportions of missing data. However, the performance may remain relatively consistent across all model sizes for both the weather and ecl datasets, even as the proportion of missing data increases from 12.5% to 50%. This stability in performance suggests that PINFDIT's architecture may achieve its optimal capacity for these imputation tasks even at smaller model sizes. Thus, the trade-off between computational resources and performance gains should be considered when selecting the appropriate model size for specific applications.
E EXPERIMENTS SETTING
E.1 BASELINES
We conduct a comprehensive comparative analysis, benchmarking PINFDiT against diverse ar- ray of leading models in the field. Our analysis extends to state-of-the-art probabilistic models, encompassing TimeGAN (Yoon et al. 2019). TimeVAR (Desai et al. 202T), Diffusion-TS (Yuan &
Qiao 2024), CSDI (Tashiro et al. (202)). TimeGrad (Rasul al. (021). TransMAF (Rasul et al. 2020). GP GP-copula (Salinas et al. 2019). and TSDiff (Kollovich et al. (022). We also evaluate against
cutting-edge deterministic models, including DLinear (Zeng et al. 2023), GPT-2 (Zhou et al. 2023). TimesNet (Wu et al.) 2023). PatchTST (Nie et al. 2023), ET (Woo et al., (022), FEDformer (Zhou et al., 2022), LighTTS (Zhang et al., 2022), Autoformer (Wu et al. 2021). and Anomaly Transformer [Xu et al. 2021), LatentODE and LubbaoCD
Rubanova et al. 2019), etc. Furthermore, we include comparisons with recent forecasting foundation models, such as TEMPO (Cao et al. 2023b), Moirai (Woo et al. 2024b), and LagLLama (Rasul et al.) 2023). This extensive comparison allows us to thoroughly evaluate PINFDiT's performance across a wide spectrum of methodologies and architectures in time series modeling.
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Published as a conference paper at ICLR 2026
E.2 DATASETS
Please
refer
to Chronos
for our pre-trained dataset and the evaluation dataset are listed as
follows:
1. The ETT (Electricity Transformer Temperature) datasets (Zhou et al. 2021 include elec- tricity load data at various resolutions (ETTh & ETTm) from two different electricity stations.
2. The Weather dataset (Zhou et al. 2021 comprises 21 meteorological indicators collected in Germany over the span of one year.
3. The Electricity (ECL, Electricity Consuming Load) Zhou al. 2021 dataset provides information on electricity consumption.
4. The SMD dataset (Su et al., 2019) includes multivariate time-series data collected from server machines in data center. It typicallyy contains metrics such CPU usage, memory usage, and disk activity.
5. The PSM dataset (Abdulaal al. sensor data from industrial machines. It often contains readings such as temperature, pressure, and vibration over time.
2021 is used for predictive maintenance and includes
6. The MSL dataset (Hundman et al. 2018 comes from the Mars Science Laboratory mission, specifically the Curiosity rover. It includes telemetry data from the rover's sensors and systems.
7. The SWaT dataset (Mathur & Tippenhauer 2016)
originates from scaled-down water treatment testbed designed to reflect a real-world water treatment process. It includes sensor and actuator data collected over time.
8. The SMAP dataset (Hundman al., 2018) comes from NASA's Soil Moisture Active Passive (SMAP) mission, which measures soil moisture and freeze/thaw state. includes time-series data from multiple sensors aboard the SMAP satellite.
9. The Sine dataset (Yoon et al.
2019) is synthetically generated by sinusoidal waves.
10. The Air Quality dataset (Yi et al. 2016) I contains hourly averaged readings from five metal oxide chemical sensors integrated into an Air Quality Chemical Multisensor Device. This device was positioned at road level in a highly polluted area of an Italian city. Data were collected from March 2004 to February 2005, making it the longest freely available record of on-field air quality chemical sensor responses.
11. The Stock dataset (Yoon et al. 2019 contains daily historical Google stocks data from 2004 2019.
12. The UCI Appliances Energy prediction dataset (Yoon et al. 2019 consists of multivariate,
continuous-valued measurements including numerous temporal features measured at close intervals.
13. The Weather 2 dataset (Godahewa et al. 202 The Weather 2 dataset comprises hourly climate TSD collected near Monash University, Clayton, Victoria, Australia, from Jamuary 2010 to May 2021. It includes series for temperature, dewpoint temperature, wind speed, mean sea level pressure, relative humidity, surface solar radiation, surface thermal radiation, and total cloud cover.
14. The PhysioNet dataset (Silva et al. 2012 contains clinical time series data from 12,000 ICU patients, each with 42 vital variables recorded over 48 hours with naturally missing values. Patients are evenly divided into three groups of 4,000 each. For benchmarking
ETT: BTT.tr/r///aatua..ooo///ooooo//r
"The h//aa///aaa///////////////////ar
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Published as a conference paper at ICLR 2026
purposes, select out of 42 variables. To address varying scales, we apply standard normalization, resulting in features with zero mean and unit variance.
15. MIMIC-III (Bica et al. 2020 MIMIC-III dataset contains 5000 patient ICU records with 19 variables from the lab events table including "anion gap. albumin, bands, bicarbonate, bilirubin, creatinine, chloride, glucose, hematoerit, hemoglobin, lactate, platelet, potassium, PIT, INR, PT, sodium, BUN, WBC'. They irregularly sampled and process them following the previous works missing values.
(Bica et al. 2020
Cao et al. 2023a), which have naturally
16. NASDAQ: NASDAQ Top 10 Stocks dataset comprises time series data for the ten largest companies by market capitalization listed on the NASDAQ stock exchange. The dataset includes daily and 5-day price data for each stock from 2014-2024, offering two temporal resolutions for comprehensive analysis. We predict the close prices of each company in the multi-resolution forecasting task.
17. Monash dataset archive (Godahewa et al. 2021b): The Monash repository contains 30 datasets, including publicly available time series datasets in various formats and those curated by us. Many datasets have different versions based on frequency and the inclusion of missing values. We use their multivariate time series version for pre-training and evaluation (specified needed).
Table 23: Dataset details
Dataset
Domain
Length
Dimension
Frequency
ETTh
Energy
17420
7
1 hour 15 min min
ETTm
Energy
69680
7
Weather
Nature
52696
21
Electricity
Energy
26304
1 hour
Air Quality
Nature
9357
13
1 hour
Sine
Synthetic
10000
5
N/A
Stock
Finance
3685
6
day
Energy
Energy
19745
28
min
MSI.
Space
132046
min
PSM
Cloud
220322
25
min
SMAP
Space
562800
25
min
SMD
Cloud
1416825
38
min
SWaT
Energy
944920
second
Requests Minute
Cloud
64800
10
min
Function Delay Minute
Cloud
64800
10
min
Platform Delay Minute
Cloud
64800
10
min
Memory Usage Minute
Cloud
64800
10
min
CPU Limit Minute
Cloud
64800
10
min
Memory Limit Minute
Cloud
64800
10
min
Instances Minute
Cloud
64800
10
min
Weather 2
Climate
3001
695
day
PEMS
Traffic
4320
852
1 hour
PhysioNet(b)
Health Care
7
Irregular
PhysioNct((b)
Hcalth Care
7
Trregular
PhysioNet(e)
Health Care
7
Irregular
MIMIC-III
Health Care
19
day
NASDAQ
Finance
2516
20
Multiresolution
E.3 METRICS
MAE
describes the mean absolute error that measures the absolute difference between ground truth and prediction.
MAE
(61)
1=1
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Published as a conference paper at ICLR 2026
MSE
describes the mean squared difference between ground truth and prediction.
MSE Σ i=1
(t Ds)²
(62)
RMSE
is the sqaure of MSE.
RMSE
ys 9.)²
(63)
Discriminative score Following TimeGAN, we train a post-hoc time-series classification model (by optimizing 2-layer LSTM) distinguish between sequences from the original and generated datasets. First, each original sequence is labeled real, and each generated sequence is labeled not real. Then, an off-the-shelf (RNN) classifier trained distinguish between the classes standard supervised task. We then report the classification еπто on the held-out test set.
Predictive Score Following TimeGAN, we train a post-hoc sequence- preencccppriitioonnn model (by op- timizing a 2-layer LSTM) to predict next-step temporal vectors over each input sequence. Then, we evaluate the trained model on the original dataset. Performance is measured in terms of the mean absolute error (MAE); for event-based data, the MAE is computed as the absolute value estimated probability that the event occured.
Computations of CRPS We explain the definition and calculation of the CRPS metric. The continuous ranked probability score (CRPS) assesses how well an estimated probability distribution F aligns with an observation It is defined the integral of the quantile loss A, (g. 2) (a (2 q) over all quantile levels a E [0, 1]:
CRPS(F-1,x)) [ 2Aa da
(64)
where represents the indicator function. We then calculated quantile losses for quantile levels discretized in 0.05 increments. Thus, we approximated CRPS as follows:
CRPS(F .x)
Σ
2..oo.ssF=11 (i-0005),,
(655
19
Next, we computed the normalized average CRPS for all features and time steps:
Σx, CRPS(FE) xK.)) Σx, [2a,]|
CRPS Score
(66)
where l and I denote the features and time steps of the imputation targets, respectively. The lower the CRPS, the more accurate the model, i.c., the closer the predicted probability is to the observed outcome.
Computations of CRPS sum CRPS sum measures CRPS for the distribution F' of the sum of all K features, calculated by:
Σ, CRPS(F-1, E 2k.)
CRPS sum Score
(67)
Σx, [
where Ee Fk.t is the total of the forecasting targets for all features at time point I.
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Published as a conference paper at ICLR 2026
Precision Precision measures the accuracy of positive predictions made by a model. It is defined as the ratio true positives (TP) to the total number of predicted positives, which includes both true positives and false positives (FP). Mathematically, precision is expressed as:
TP Precision TP FP
(68)
Recall Recall, also known sensitivity, measures model's ability to correctly identify true positive instances. It is calculated as the ratio of true positives (TP) to the sum of true positives and false negatives (FN). In the context of anomaly detection, failing to detect an anomalous timestamp can have serious consequences, making recall a critical metric. Mathematically, recall is defined as:
TP
Recall TP FN
(69)
F1-score The Fi-score is a balanced measure of model performance that combines Recall and Precision. It calculated as the harmonic mean of these two metrics, giving equal importance both. This score effectively captures the trade-off between Recall and Precision, penalizing significant disparities between them. By providing a single, comprehensive metric, the F1-score offers a more holistic view of a model's effectiveness, particularly useful when dealing with imbalanced datasets.
Fl 2 Precision Recall
(70)
Precision + Recall
F PINFDIT PARADIGM ON TRAINING AND INFERENCE
Borader Impact and Position of PINFDIT PINEDIT bridges the critical gap between specialized deep learning baselines and the ideal of a universal time series foundation model. While pioneering works such as CSDI have successfully leveraged transformer architectures for probabilistic imputation, they generally rely on random masking strategies and concatenation-base design choices often constrain their effectiveness when generalizing to diverse downstream tasks like long-horinn Long-horizon forecasting physically consistent generation. In contrast, PINFDIT embodies the characteristics of a true foundation model through a balanced design that fuses data-driven versatility with domain-specific rigor. Its foundation model status is established through three distinctive advancements: (1) Unified Representation Learning: Unlike predecessors that focus primarily on random imputation masks, our Time Series Mask Unit (TSMU) integrates structured masking strategies (block, stride, and reconstruction). This allows the model to learn robust temporal dependencies across varying channel sizes and sequence lengths without requiring -ase-spec architecture changes. (2) Task-Agnostic Versatility: By employing Adaptive Layer Normalization (AdaI.N) for deep conditional fusion, PINFDIT creates a highly responsive generative framework capable of seamlessly transitioning between forecasting, imputation, anomaly detection, and synthetic generation within single pre-trained model. (3) Physics-Informe
conditioning. These
Reliability: Uniquely, we introduce physics-informed sampling mechanism via Langevin dynamics. This energy- approach allows for the modular injection of domain knowledge (e.g., conservation laws) during inference, ensuring scientific plausibility without the need for expensive retraining.
Although technical challenges such as extreme sequence length scaling remain, this combination of architectural flexibility, structured pre-training, and post-hoc physical consistency positions PINFDiT as practical "proto-foundatio" " model, advancing the field from single-task solvers toward universal, scientifically grounded time series modeling.
Synergy between Architecture and Diffusion on Irregular Data. We attribute our superior performance irregular and missing data not merely the use of diffusion framework, but to the specific synergy between the diffusion objective and our architectural innovations. While diffusion models inherently support flexible conditioning, it is Time Series Mask Unit (TSMU) and conditioning strategies that fully exploit this capability.
Specifically, our ablation studies demonstrate that:
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Published as a conference paper at ICLR 2026
AdaLN Conditioning: Replacing standard addition cross atossaattention with Adaptive Layer Normalization (AdaLN) significantly lowers error, proving essential for robust temporal modeling.
Stride Masking: The use of stride masking during training teaches the model handle non-contiguous dependencies, capability that standard U-Net architectures (like CSDI) often lack.
Our "What You See Is What You Get" embedding with temporal attention is critical; replacing it with standard Patch Tokens results in a severe performance drop.
Temporal-wise Attention:
Thus, PINFDiT succeeds where specialized ODE/CDE models struggle because the diffusion pro- cess provides the generative flexibility, while our specialized transformer architecture provides the structural capacity to model complex, irregular temporal dynamics.
Training details Similar the previous DiT work (Peebles & Xie 2022 PINFDiT is available in four sizes: small (S. 33M parameters), big (B, 130M parameters), large (L. 460M parameters), and extra large (XL, 680M parameters). A comprehensive comparison in Table 25 shows PINFDiT's expanded task coverage relative to existing general-purpose time series models, including anomaly detection, imputation and data generation. with a learning rate of 0.0001 and the loss function is from Equantion ??. Batch sizes of 256 or 512 were employed, depending on model size. The ideal epoch to convergence over 100 as the complexity of training data, but we choose to use the earlier checkpoint for the case of downstream purpose anomaly detection and synthetic generation because the two tasks are very datasct-specific and do not necessarily benefit from learning distributions beyond the target dataset. In practice, the maximum channel number (K mazz) was set to 20-40, with a maximum sequence length of 198, unless otherwise specified. All experiments were conducted on NVIDIA A100 GPUs with 40G GPU memory. Importantly, our zero-shot foundation model was trained on Chronos dataset without exposure any data from the evaluated downstream tasks datasets. The results of zero-shot section are derived from a single pre-trained checkpoint, evaluated with or without fine-tuning based on the settings. The only exception is the long-term zero-shot experiments, which require extended sequence inputs while still utilizing the same pre-training dataset. To facilitate reproducibility and further research, we will release the pre-trained checkpoint.
our training process, we utilized the Adam optimizer
Inference In the finetuning and inference stage, the choice of mask is tailored to align with the specific requirements of the user. This flexibility allows PINFDiT to apply the most ap- propriate masking strategy based on the context of the task and application. During infer- ence, while the mask type and parameters are fixed for a given task to ensure consistency, PINFDiT's generative task architecture allows for flexible transformation of various downstream tasks. This adaptability enables us to address wide range of time series challenges within unified framework. Let n represent the number of samples generated for each prediction, which we set to n = 10 (n = 30 for forecasting tasks) in our experimental setup at inference time. We use the median of these n predictions as the final prediction, providing the added benefit of obtaining a confidence interval for PINFDiT's predictions. To prevent channel padding from affecting the generated samples, we mask out the invalid channels during sampling at each dif- fusion timestep so that PINFDiT does not falsely treat the information in the non-valid channels as meaningful information. Padding is applied at the beginning of the temporal dimension to ensure that the most relevant information remains at the end, thereby mitigating the effect of padding. We have included inference time comparisons for single-sample generation, where PINFDiT demon- strates superior computational efficiency, requiring only second for single-sample generation, making it more practical for real-world applications. Regarding the Langevin refinement introduces minimal computational overhead of approximately 0.003s per iteration. For a standard setting with K 10 refinement steps, the total inference time for batch of 50 samples is approximately 1. 1.05s. This efficiency stems from the optimized gradient computation,
Table 24: Comparison of inference times for single- sample generation.
Model Diffusion-TS
Inference Time (mm:ss)
00:06
CSDI
00:02
PINFDIT
00:01
module, the
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Published as a conference paper at ICLR 2026
which only requires O(N T' operations (where N 40 spatial and T 192 temporal points), making the physics guidance practical for real-time applications.
Table 25: A comparable analysis of representative general purposes time series models
Model Lap-
Parameter Size
Model Architecture
Channe Setting Univarioee
Task Type Forressting
Pretrain Datasef
Dada Slzt
Transfomer
Monadh (rabhowa
N01
300 Million Time Points
Tranfornas
Caivarizte
LOTSA Wio
2024
27 Rillian
3115
PINTDIT
E:460IM
Tranformer
Chronce
Billion Time
Generation
Limitations and Future Work. Despite PINFDiT's strong performance, several limitations remain. First, our model inherits computational constraints common to diffusion models, requiring multiple sampling steps that increase inference time compared to single-pass methods. This presents challenges for real-time applications requiring instant predictions. Second, while PINFDiT effectively handles missing values and irregular sampling, extremely sparse time series with large gaps may still present difficulties, particularly when physical constraints operate across these gaps. Third, our current physics injection method works best with explicit differential equations; incorporating implicit physical knowledge constraints expressed through other mathematical formulations remains challenging. Fourth, like most foundation models, PINEDiT's performance depends on the diversity and quality of the pretraining data, potentially limiting generalization to entirely novel physical systems not represented in the training distribution. Finally, while we demonstrate strong performance various scientific time series benchmarks, the evaluation of extremely long-term dependencies (spanning thousands of time steps) and the incorporation of multi-modal data (e.g., simultaneous text descriptions and time series measurements) represent important directions for future work. Addressing these limitations will be crucial for developing truly universal foundation models for scientific time series analysis.
G USE OF LARGE LANGUAGE MODELS (LLMs)
We utilized large language models (LLMs) as a general-purpose writing assistant to slightly polish the manuscript. The LLM's role was strictly limited to improving grammar, clarity, and style. The LLM did not contribute to the research ideation, experimental design or the core scientific conclusions presented in this paper. Therefore, the LLM is not regarded as contributor to this work.
47


Reviews:
[{"review_id": "CoBvOJPyid", "reviewer": "Reviewer_7Sa8", "summary": "The paper proposes using diffusion for time series modeling. The model is a specially designed transformer architecture that can deal with multivariate time series with missing values. Additionally, authors apply physical law adherence through the direct optimization-like procedure in the sampling. The experiments are extensive and show good results on many different setups.", "questions": "- How do results in 4.1 compare to more conventional solvers? Is there any benefit to your method?\n- When should one use your method? To improve the accuracy or to learn the unknown dynamics?\n- Can you comment on the runtime, both for the training and inference? Especially the conventional diffusion sampling compared to the physics guided part.\n- Can Algorithm 1 be augmented to have guidance on the function K in the T diffusion steps instead of the second loop?\n- Do you have an explanation why the model performs better on irregular time series data compared to the some of the specialized models, which build around irregularity?\n- Can you speculate if the reason is the diffusion or the architecture? This can be an ablation study.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The physics injections in the sampling of diffusion is a very natural extension and can guarantee the model behaves as intended. This is very powerful tool as virtually any constraint can be observed. In this work, authors apply PDE constraint and additionally provide theoretical results for this specific scenario. The architecture is sound and appears to outperform competitors. The paper is clearly written and easy to follow. There are many different experimental setups. In all of them authors demonstrate good results. The appendix is extensive and contains additional theoretical proofs and experiments.", "weaknesses": "The main contribution of the paper seems to be the physics informed sampling. This part is under-explored and the authors only test a couple of datasets with non-physics models. More focus on this part would be appreciated given the title and general theme.\n\nThe rest of the experiments seem to not have any relation to the physics informed part. In my understanding there is no physics injection done here. The contribution is the architecture, which is interesting but not overly novel. Can the authors comment on what part of the architecture they believe is the most important in outperforming existing diffusion models like CSDI.\n\nMajor formatting issue: font size is reduced after eq 6.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "me4QaX9Gn3", "reviewer": "Reviewer_MmYJ", "summary": "The paper proposes PINFDIT, a diffusion model for (conditional) time series generation, adhering to physical constraints via a physical injection post-sampling via Langevin dynamics. In their empirical evaluation, the model outperforms previous approaches.", "questions": "See weaknesses and:\n\n- What are the differences between the diffusion model (without the physical injection) and previous ones?\n- Can the physical injection also be done during generation via guidance rather than as a post-processing step?\n- Can the Langevin dynamics, i.e., the physical injection, also be applied to autoregressive models? For example, Chronos?\n\nI like the idea of the physical injection, but the current state of the paper has certain uncertainties, especially differences to previous approaches. If these are addressed, I am willing to increase my score.", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "- The paper introduces a physical injection during/post inference. The solution via Langevin dynamics is elegant and comes with a theoretical justification.\n- Experiments are conducted across different physical systems and demonstrate strong performance. \n- In general, the empirical evaluation is thorough and convincing. It also compares to recent foundation models.", "weaknesses": "- A background section is missing. Certain things as the ELBO in Eq. 2 and the diffusion sampling in Eq. 1 should be moved to a background section and be explained accordingly.\n- Contributions and advancements over previous works are not entirely clear to me. E.g., L94-98, how is the model different from other diffusion-based models using a transformer backbone, e.g., CSDI [1]?\n- Self-supervised learning is mentioned in the methodology, but it is unclear how it differentiates from CSDI [1].\n- The notation is inconsistent. Sometimes the subscript for $x$ denotes the diffusion time (e.g., Eq. 1), sometimes it denotes the iteration (e.g., Eq. 8). The temporal time is also denoted with $t$ (e.g., Eq. 4 and 5).\n\nMinors:\n\n- L283: Sentence is broken\n- L284: likelihood -> the likelihood\n- Different styles to denote target *tar*, context *con*, and $\\mathbf{x}$ across the paper.\n\n[1] **Tashiro, Y., Song, J., Song, Y., & Ermon, S.** (2021). Csdi: Conditional score-based diffusion models for probabilistic time series imputation. Advances in neural information processing systems, 34, 24804-24816.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "pBnz3iDCL5", "reviewer": "Reviewer_m5ac", "summary": "The paper introduces PINFDiT, a diffusion-transformer for time series that injects physics at inference by combining a learned conditonal model with an energy term that penalized PDE/physics residuals $K(x;\\mathcal{F})$. Sampling targets the Boltzmann-type density $\\propto \\exp{-K(x;\\mathcal{F})} - \\alpha \\log p_\\theta (x|x_\\text{con})$ via Langevin/gradient updates, so no retraining is required. Architecturally, the model uses temporal-wise attention and flexible masking to handle irregular and missing data. Experiments span PDE simulators, ERA5 temperature forecasting, long-horizon \"practical\" benchmarks with uncertain metrics, and imputation/anomaly detection and generation, where they report consistent gains over baselines and ablations.", "questions": "### On inference cost reporting\nThe proposed \"physics-injection\" methods is an iterative Langevin MC loop that runs for $k$ stops. This is an additive computational cost applied at inference time for every sample generated, on top of the $T$-step diffusion sampling processes.\n- Critical omission. The paper never states the value of $k$ used in any of its experiments. This is an important hyper parameter that directly determines the inference cost. \n- In Table 19, the authors claim an inference time of 1 second per sample. Is this the time for the full PINFDIT, or for the base model?\n\nFor other questions/suggestion please see weaknesses.", "limitations": "", "rating": 2, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "- The most significant strength of this paper is its comprehensive evaluation. The authors conduct an extensive four-part experimental validation covering PDE simulations, real weather ERA5 data, multiple generative tasks (imputation, anomaly detection, etc) and zero shot generalization. The comparison agains a wide range of strong, modern baseline is thorough.\n- The paper tackles a highly relevant and challenging goal of creating a single, general-purpose model for scientific time series. This domain is critical, and the paper's explicit focus on imperfect data, physical constraints, and limited samples is well-motivated.\n- While individual components are not novel, making them work robustly across domains **at scale** is non-trivial. The idea of constrained generation without retraining is practically appealing. The generative and general-purpose capabilities are promising.\n- The foundation is principled, provides rigor and clarity. \n- The paper reports state-of-the-art or highly competitive results on the experiments.", "weaknesses": "### Overstatement of Novelty; core \"theory\" is a standard energy composition\nThe paper presents its core \"physics informed sampling\" framework (Section 3.2) as a new development, complete with theoretical backing. \n- The claim is \"develop a model-editing-free physics knowledge injection framework\" and present \"Theorem 3.1\" as a closed-form solution for their optimization problem (Eq 6).\n- The reality: this entire \"model-editing-free physics knowledge injection framework\" is a well estimated, standard technique\n    -  Theorem 3.1 is a foundational principle of statistical mechanics and Energy-Based Models (EBMs). It simply states that the optimal distribution for a composed energy function is a Boltzmann distribution (a product of experts, PoE, established in 1999). The proof in Appendix A.3 is a standard Lagrange multiplier derivation yielding Bayes theorem. \n    - Algorithm 1 is standard Langevin MC applied to the composed energy function ($\\nabla\\log q^\\ast = \\nabla K + \\alpha \\nabla\\log p_\\theta$) . Theorem 3.2 restates standard LMC results under strong assumptions, which is quite general and provides no insight for this specific problem.\n\nBy presenting these foundational, decades-old principles of EBMs and its sampling algorithm as theorems derived for this paper with new names like \"model-free physics-guided correction\", \"calibrated Langevin dynamics\" and by failing to to cite or acknowledge the extensive prior art on Product of Experts, EBM composition, or the vast field of generative models for inverse problems that uses this exact mechanism, the authors overstate the novelty of their core contribution relative to well-established energy-guided/compositional sampling. It would be great that the authors can clarify the distinction and situate this work relative to relevant prior art. \n\n\n### Lack of Rigorous Statistical Validation\nThe paper fails to report any measure of experimental variance (e.g., standard deviations or confidence intervals over multiple runs) for nearly all of its main results. This is a critical omission, especially for a stochastic model. This is problematic when assessing the paper's central claims, where the reported differences are often small:\n- weakens the core contribution: In the crucial ablation study (Table 7), the performance difference between the full PINFDIT model (CRPSsum 0.424 on Solar) and the model \"w/o Phys\" (0.445) is marginal.  The paper compares the full PINFDIT (with Langevin correction) against PINFDIT DDPM (without it) to prove the value of the physics-injection. The differences are minuscule:\n    - Navier-Stokes (RMSE): 0.0030 (PINFDIT) vs. 0.0031 (DDPM)\n    - Advection (RMSE): 0.0035 (PINFDIT) vs. 0.0036 (DDPM) \n    - Diffusion-Sorption (RMSE): 0.0049 (PINFDIT) vs. 0.0050 (DDPM)\n\nWithout error bars, it's impossible to conclude that the Langevin correction (a key contribution) provides any significant benefit; This directly undermines the paper's entire narrative about the practical value of its main \"physics-informed\" contribution. \n- Obscures SOTA Comparisons:The paper claims SOTA performance, but many \"wins\" are tiny and could be statistical noise.\n\n    - Practical Forecasting (Table 3): On PhysioNet(a), PINFDIT achieves an MAE of 0.616 vs. CSDI's 0.620. On PhysioNet(c), it's 0.543 (CRPS) vs. CSDI's 0.548. These sub-1% differences are meaningless without a measure of variance.\n\n    - Anomaly Detection (Table 14): On the PSM dataset, PINFDIT scores an F1 of 97.57, while TimesNet gets 97.34 and TimeLLM gets 97.23 . A ~0.2% difference is well within any reasonable margin of error. It is misleading to claim superiority based on this.\n\nThe fact that the authors do report variance in Table 12 for the synthetic generation task (e.g., \"0.031(0.007)\") confirms they are aware of this analysis and capable of performing it. This makes its conspicuous absence from all other key results tables (Tables 1, 3, 6, 7, 9, 10, 11, 15, etc.) a more significant methodological flaw that prevents a proper assessment of the paper's claims.\n\n### Physics operator $K$ is under-specified for ERA5: mismatch to target variable.\nThis paper does state the output is 2-meter temperature prediction. And in appendix A.7, the paper lists the resolution and date splits. While the target as established, is to predicted 2 meter temperature, while in Appendix A.7, under \"Physical Constraints\", the only equation they provide is the Navier-Stokes momentum equation. This equation governs the evolution of wind velocity, not temperature. The physics of temperature near the surface is governed by a different, unmentioned equation (and in fact, t2m is a diagnostic quantity, obtained by Monin-Obukhov interpolation between skin temperature and the lowest model level, and is governed by surface energy-balance and boundary-layer physics). The paper explicitly states, \"The model's output time series is post-processed by penalizing the residuals... of the above equation\". This is physically meaningless. A temperature field does not and cannot satisfy the momentum equation. The authors have either applied a completely irrelevant physical constrained, or critically failed to describe their experiment. In either case, this is a major flaw. This gap makes their \"real-world\" physics-informed experiment unsound and irreproducible as described. \n\n### Limited applicability of the \"general-purpose\" physics module.\nThe proposed \"physics-injection\" model is far less \"general-purpose\" than the base model it is applied to, limiting its practical utility. The methods is entirely reliant on the existence of a known, explicit and differentiable residual function. The authors concede this point in their own limitation section.\n\n### On experiments\nThe paper's claims are undermined by some flaws in its experimental design.\n- Post-hoc refinement vs. no analogue in baselines. PINFDiT uses a second-stage Langevin refinement after the diffusion sampler. Table 1 then compares against NeuralODE/NeuralCDE and other models without an analogous inference-time correction. So it is comparing (PINFDiT + k steps physics correction) against (Neural ODE + no correction). It only proves that adding a post-hoc, physics-based gradient correction step improves the final results, which is trivial and obvious conclusion. A methodologically sound experiment  would have been to apply the same k-step gradient based physics correction to the outputs of all baseline models, and then compare the final errors. \n\n- Inappropriate baselines. In Table 1, the paper compares its forecasting model against SNPE and LFBC. Theses are Simulation-Based Inference (SBI) methods designed for parameter posterior estimation, not time-series forecasting. Their inclusion here is confusing and does not directly test the claimed inference-time energy guidance against the most comparable alternatives. \n- Mixing deterministic and probabilistic baselines on CRPS. In Table 1 and 3, the paper evaluates deterministic, point-estimate models (e.g., DLinear, PatchTST) using probabilistic metrics like CRPS. This is a nonsensical comparison. These models do not natively produce predictive distributions. Also. the paper does not explain how this was calculated.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Pu33yhAQoA", "reviewer": "Reviewer_EgRK", "summary": "This paper proposes PINFDIT, a novel framework aiming to be a general-purpose model for time series analysis. It combines a Transformer architecture with a Diffusion framework to perform a variety of tasks, including forecasting, imputation, anomaly detection, and synthetic data generation. The core contributions of the paper are (1) a comprehensive masking strategy to handle imperfect time series data (missing values, multi-resolution) and (2) a novel \"energy-based physics-informed sampling\" technique that allows for the injection of physical laws at the inference stage without retraining or architectural modification.\n\nThis work tackles a very ambitious and important problem. The attempt to unify a pre-trained, general-purpose model with domain-specific physical knowledge is highly timely. The use of calibrated Langevin dynamics for correction at inference time is a practical and clever idea that could overcome the limitations of existing approaches like PINNs.\n\nHowever, there are several significant weaknesses regarding the paper's core claims, experimental fairness, and methodological clarity. Without addressing these, it is difficult to evaluate the generality and true contribution of the proposed framework.", "questions": "1.  **(Re: W1)** Could you please provide a detailed explanation of how the partial differential terms and the gradient $\\nabla K$ in Eq. (5) and Eq. (8) are computed for discrete time series data $x^{tar}$?\n2.  **(Re: W3)** What was the specific reason for \"bypassing pretraining\" in the anomaly detection task? Does this imply that the pre-trained general-purpose model may be unsuitable for certain downstream tasks?\n3.  **(Re: W3)** For a fair comparison in the imputation experiments (Tables 4/15), can you provide results for PINFDIT trained from scratch (without pre-training)?\n4.  **(Re: W4)** Can you show what happens to performance when the physics-injection module (Alg. 1, lines 6-8) is applied with an intentionally incorrect physical law, or to a non-physical dataset like NASDAQ?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1.  **Novel and Practical Physics-Injection Method:** Unlike traditional Physics-Informed Machine Learning (e.g., PINNs) which requires including PDE residuals in the training loss and retraining for specific problems, the proposed \"model-editing-free\" inference-time correction is highly practical. The ability to maintain the flexibility of a pre-trained general model while \"plugging in\" domain knowledge as needed is a major advantage.\n2.  **Comprehensive Problem Definition and Task Scope:** The paper has an ambitious goal of solving a wide range of time series tasks (forecasting, imputation, anomaly detection, generation) with a single model. It is also commendable that it directly addresses real-world data complexities such as missing values, multi-resolution sampling, and irregular intervals.\n3.  **Extensive Experimental Validation:** The authors have conducted a vast number of experiments to demonstrate their model's performance. This includes PDE simulator-based forecasting, practical forecasting on real-world data (climate, healthcare, finance), generative and imputation tasks, zero-shot performance, and detailed ablation studies. Achieving state-of-the-art (SOTA) performance on many benchmarks is impressive.", "weaknesses": "1.  **Ambiguity in the Implementation of Physics Injection:**\n    * The concrete implementation of the physics-injection mechanism, one of the paper's core contributions, is critically unclear.\n    * Physical laws are quantified by an energy function $K(x^{tar};F)$ (the squared PDE residual).\n    * For correction during inference, the gradient of this energy function ($\\nabla K(x_{j}^{tar};F)$) is required (see Algorithm 1).\n    * **Critical Question:** $x^{tar}$ is discrete time series data. How are partial differential terms like $\\frac{\\partial x^{tar}}{\\partial t}$ and $\\frac{\\partial x^{tar}}{\\partial u_{i}}$ computed? If finite differences were used, there is no discussion of the associated discretization error, stability, or sensitivity to the choice of scheme. This information is essential for reproducing the methodology and verifying its validity.\n    * The authors claim they do not require a \"differentiable simulator\", but it appears they *do* require a \"differentiable (physical) residual function.\" The distinction and constraints are not adequately explained.\n\n2.  **Disconnect in \"General-Purpose\" and \"Foundation Model\" Claims:**\n    * The paper positions PINFDIT as a \"proto-foundation model\".\n    * However, the model's operation contradicts this claim. The model's pre-training (using the Chronos dataset) is entirely physics-agnostic and learns only statistical patterns.\n    * The physics injection is merely a post-hoc correction step (Algorithm 1, lines 6-8) that operates *outside* the pre-trained model during inference. This is less a \"Physics-Informed\" Transformer and more a \"General-Purpose Transformer\" combined with a \"Physics-Based Corrector Plugin.\" While this modularity is a strength, the title \"Physics-Informed\" is misleading as the model itself does not *learn* or *internalize* the physics.\n\n3.  **Unfair Experimental Comparisons and Lack of Consistency:**\n    * **Imputation (Tables 4 & 15):** The authors state, \"All baseline models are trained in a full-shot setting, while PINFDIT leverages a pre-trained foundation model, fine-tuning it on realistic datasets\". This is a **clearly unfair comparison**. PINFDIT benefits from pre-training on massive external data (Chronos, ~5B time points), while baselines like PatchTST and TimesNet are trained from scratch only on the task data (e.g., ETTh1). It is impossible to distinguish if PINFDIT's superiority comes from its architecture or simply from pre-training. A fair comparison would require reporting results for PINFDIT trained from scratch on the same data.\n    * **Anomaly Detection (Table 14):** The authors state they \"opted to bypass pretraining\" for this task and introduced an additional pre-processing step (Spectral Residue). This severely undermines the core narrative of a single foundation model for all tasks. If the pre-trained model is actually detrimental for anomaly detection (as it \"may inadvertently overfit by reconstructing anomalies\"), this clearly exposes a limitation of the proposed general-purpose model.\n\n4.  **Lack of Robustness Analysis for the Physics-Injection Plugin:**\n    * The paper only presents positive cases where physics injection improves performance (e.g., PDE simulations, ERA5 climate forecasting).\n    * There are no failure-case or sensitivity analyses. What happens if an incorrect physical law is injected (e.g., applying Burgers' equation to non-fluid data)? What happens if it's applied to data with no physical basis (e.g., the NASDAQ financial dataset)? Does performance degrade? Proving the validity of this modular approach requires such robustness checks.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)