Title: TARFVAE: Efficient One-Step Generative Time Series Forecasting via TARFLOW based VAE
Venue: NeurIPS 2025
Accepted: yes
Keywords: time series forecasting

generative model

normalizing flow

variational autoencoder
Abstract: Time series data is ubiquitous, with forecasting applications spanning from finance to healthcare. Beyond popular deterministic methods, generative models are gaining attention due to advancements in areas like image synthesis and video generation, as well as their inherent ability to provide probabilistic predictions. However, existing generative approaches mostly involve recurrent generative operations or repeated denoising steps, making the prediction laborious, particularly for long-term forecasting. Most of them only conduct experiments for relatively short-term forecasting, with limited comparison to deterministic methods in long-term forecasting, leaving their practical advantages unclear. This paper presents TARFVAE, a novel generative framework that combines the Transformer-based autoregressive flow (TARFLOW) and variational autoencoder (VAE) for efficient one-step generative time series forecasting. Inspired by the rethinking that complex architectures for extracting time series representations might not be necessary, we add a flow module, TARFLOW, to VAE to promote spontaneous learning of latent variables that benefit predictions. TARFLOW enhances VAE's posterior estimation by breaking the Gaussian assumption, thereby enabling a more informative latent space. TARFVAE uses only the forward process of TARFLOW, avoiding autoregressive inverse operations and thus ensuring fast generation. During generation, it samples from the prior latent space and directly generates full-horizon forecasts via the VAE decoder. With simple MLP modules, TARFVAE achieves superior performance over state-of-the-art deterministic and generative models across different forecast horizons on benchmark datasets while maintaining efficient prediction speed, demonstrating its effectiveness as an efficient and powerful solution for generative time series forecasting. Our code is available at https://github.com/Gavine77/TARFVAE.

TARFVAE: Efficient One-Step Generative Time Series Forecasting via TARFLOW based VAE

Abstract

Time series data is ubiquitous, with forecasting applications spanning from finance to healthcare. Beyond popular deterministic methods, generative models are gaining attention due to advancements in areas like image synthesis and video generation, as well as their inherent ability to provide probabilistic predictions. However, existing generative approaches mostly involve recurrent generative operations or repeated denoising steps, making the prediction laborious, particularly for long-term forecasting. Most of them only conduct experiments for relatively short-term forecasting, with limited comparison to deterministic methods in long-term forecasting, leaving their practical advantages unclear. This paper presents TARFV AE, a novel generative framework that combines the Transformer-based autoregressive flow (TARFLOW) and variational autoencoder (VAE) for efficient one-step generative time series forecasting. Inspired by the rethinking that complex architectures for extracting time series representations might not be necessary, we add a flow module, TARFLOW, to VAE to promote spontaneous learning of latent variables that benefit predictions. TARFLOW enhances VAE's posterior estimation by breaking the Gaussian assumption, thereby enabling a more informative latent space. TARFVAE uses only the forward process of TARFLOW, avoiding autoregressive inverse operations and thus ensuring fast generation. During generation, it samples from the prior latent space and directly generates full-horizon forecasts via the V AE decoder. With simple MLP modules, TARFVAE achieves superior performance over stateof-the-art deterministic and generative models across different forecast horizons on benchmark datasets while maintaining efficient prediction speed, demonstrating its effectiveness as an efficient and powerful solution for generative time series forecasting. Our code is available at https://github.com/Gavine77/TARFVAE.

Pengbo Wei

Meituan

Beijing, China

Chen Chen

Meituan

1 Introduction

In various application fields such as transportation planning[1], healthcare[2] and inventory management[3], long-term time series forecasting (LTSF) and uncertainty quantification are of vital importance. The former provides a solid basis for long-term decision-making by offering reliable prediction, while the latter delivers more comprehensive estimates, enabling robust decisionmaking through accounting for potential variability. Recently, deep learning has achieved remarkable success in these fields through capturing complex time series patterns[4-9] and generating reliable uncertainty estimates[10-13].

In the realm of long-term forecasting, various strategies have been proposed to enhance predictive performance. Most efforts have focused on Transformer-based models due to their proven capability in capturing long-term dependencies via attention mechanisms[6]. Numerous studies aim to reduce the computational complexity of attention[6-8] and improve information extraction, while others apply vanilla Transformers to mine intra-channel or inter-channel relationships[4, 5]. However, some studies[14, 15] argue that sophisticated designs might not be necessary and suggest simple models could deliver results comparable to Transformer-based models. This has spurred exploration into Linear-based or MLP-based methods that aim at more efficient utilization of historical information during training to improve prediction performance[9, 16, 17].

Current LTSF methods are predominantly deterministic, producing point estimates without systematically addressing uncertainty quantification. This oversight poses substantial risks to operational robustness, particularly given the intrinsic positive correlation between uncertainty magnitude and forecast horizon length. To achieve probabilistic forecasting, earlier models like DeepAR[12] employ recurrent neural networks (RNNs) to estimate parameters of prespecified distributions (e.g., Gaussian) for each timestep. However, such methods impose restrictive parametric assumptions that may not capture complex temporal dynamics. Subsequent advancements have explored generative frameworks, including variational autoencoders (VAEs)[18] and diffusion models[10, 13], building upon their demonstrated success in adjacent domains[19-21]. While diffusion models achieve state-of-the-art sample quality, their iterative denoising process incurs substantial computational overhead during inference. Recent attempts to accelerate generation[11, 13, 22] remain fundamentally constrained by the multi-step generation paradigm. By contrast, conventional V AE[23] enables one-step generation by decoding samples from the prior latent space. However, many existing hybrid methods[24-26] introduce complex recurrent structures to capture temporal dependencies and generate predictions autoregressively, which undermine this computational advantage. Due to computational constraints, these generative models have primarily been developed and evaluated in short-term settings, raising concerns about their long-term efficacy and accuracy. Recently, Zhang et al.[27] systematically compared deterministic and probabilistic methods for long-term forecasting, whose results still highlight the need for models that can effectively address both point and probabilistic forecasting across diverse horizons.

Motivated by these considerations, we introduce TARFVAE, a novel VAE-based framework that strategically integrates two core objectives: (1) preserving the one-step generation capability of VAEs to ensure computationally efficient inference, and (2) promoting spontaneous learning of latent variables that guarantee robust probabilistic forecasting performance. TARFVAE achieves these by simply incorporating a Transformer-based autoregressive flow (TARFLOW)[28], which enables an enhanced posterior estimation. Note that the autoregressive inverse process of TARFLOW is not included so TARFVAE can perform one-step generation. Extensive experiments on eight real-world datasets demonstrate the superiority of TARFVAE over the existing state-of-the-art deterministic and generative baselines.

2 Related Work

Long-term Time Series Forecasting Transformer-based models have achieved remarkable success in LTSF through their attention mechanisms, which effectively capture long-term dependencies[68]. According to Qiu et al.[29], these models can be categorized into channel-independent and channel-dependent approaches. Initially, they aggregate channel information via linear mapping and use attention to extract temporal patterns. However, the high computational complexity of attention mechanisms poses challenges in long-term forecasting. To address this, significant efforts have focused on enhancing attention efficiency[6-8, 30]. Despite progress, channel-dependent

strategies are more vulnerable to distributional shifts[14, 31], prompting some researchers to explore channel-independent approaches, which have shown superior performance[4, 5, 14]. For example, PatchTST[4] splits historical data into patches and applies a standard Transformer to model their correlations. Meanwhile, other studies have leveraged simple Linear-based or MLP-based structures for time series modeling[14, 16, 17, 31]. For instance, SOFTS[17] introduces an MLP-based module to aggregate multiple channels and generate core representations for prediction.

Generative Models for LTSF To boost model robustness and reliability, recent studies have shifted from estimating the median or mean of time series to capturing the complete time series distribution using generative models. Various generative models, such as V AEs[23, 24], normalizing flows[32, 33] and diffusion models[10], have been applied in this field. Transformer-MAF[32] is the pioneering work in applying normalzing flows: it builds upon the Transformer architecture and utilizes Masked Autoregressive Flows (MAF)[34] for generating time series forecasts. Recently, diffusion models have gained prominence in LTSF because of their superior performance in adjacent domains[19-21]. TimeGrad[10], the first time series diffusion model, relied on an RNN to predict in an autoregressive manner, with the denoising process guided by hidden states generated by RNN. Instead of autoregressive decoding, CSDI[11] and SSSD[35] use self-supervised masking to guide the denoising process and make predictions. Methods like TimeDiff[13] and mrDiff[36] utilize different networks to incorporate historical information as conditions and employ conditional diffusion models for time series generation. However, diffusion-based models, though effective, require multiple steps to generate time series from noise, thus slowing inference. Unlike diffusion models, V AEs can efficiently generate data by directly decoding latent representations. Previous studies[18, 24-26] improve the learning of latent space by better representing historical sequences through complex recurrent or autoregressive operations, which compromise VAE's fast inference speed.

3 Preliminaries

3.1 Variational AutoEncoder

A VAE[23] is an unsupervised generative model that models the input data distribution by learning a probabilistic latent space as follows:

where x is the input data and z is its latent representations. The prior p ( z ) is normally defined as a multivariate Gaussian distribution N (0 , I ) . The posterior p ( z | x ) can be an arbitrary distribution and VAE approximates it as q ( z | x ) = N ( µ ( x ) , σ 2 ( x )) . Thus VAE learns the problem by maximizing the log-likelihood of (1) as

where (3) is the evidence lower bound (ELBO) of (2) accoring to Jensen's inequality. The first term in (5) leads to reconstruct accurate x from z , and the second term minimizes the difference between the prior and approximated posterior to prompt the latent space to capture meaningful data representations. This enables V AE to generate new samples via z ∼ p ( z ) , x ∼ p ( x | z ) .

VAE can be extended to conditional VAE by incorporating supervised labels y . The ELBO for log p ( x | y ) can be analogously derived as follows:

where the prior, approximated posterior and reconstruction process are all conditioned on y .

3.2 Normalizing Flow

Normalizing flows[33, 37] are invertible mappings f : X → Z from R D to R D . For x ∼ X and z ∼ Z , the density p X ( x ) can be expressed by

Normalizing flows have the property that the inverse x = f -1 ( z ) is easy to evaluate and computing the Jacobian determinant takes O ( D ) time. Therefore, once the model is trained, a generative model is automatically obtained via z ∼ Z , x = f -1 ( z ) . Real NVP[38] designs an affine coupling layer to meet the above two conditions:

where x can be randomly shuffled into two parts [ x 1 , x 2 ] , and s ( x 1 ) and t ( x 1 ) are learnable functions whose outputs match the shape of x 2 . The inverse of (9) is obvious and the Jacobian matrix of (9) is a lower triangular matrix. The determinant of the Jacobian is the product of the diagonal elements, so the log-determinant is

To achieve stronger nonlinearity, multiple coupling layers are composed together, creating a chain of mappings: X = Z 0 →Z 1 →Z 2 → ... →Z K = Z . The maximum likelihood estimation objective can then be written as

Recently, Zhai et al. proposed TARFLOW[28] which could achieve more efficient nonlinear mappings. TARFLOW partitions x into more parts as [ x 1 , x 2 , ..., x n ] and performs transformation following a similar rule:

where j > 1 and x <j = [ x 1 , x 2 , ..., x j -1 ] . The s ( x <j ) and t ( x <j ) are efficiently implemented by causal Transformer. Notably, the inverse of (12) becomes

where the inference of x j relies on x <j , making the inverse process autoregressive.

3.3 Time Series Forecasting

Time series forecasting uses historical time series x ∈ R C × L to predict the future values y ∈ R C × H . Here, C represents the number of variables or channels, L is the length of the lookback window, and H stands for the forecast horizon. C > 1 corresponds to a multivariate time series; otherwise, it is termed univariate.

Deterministic methods perform point estimation via a function f as ˆ y = f ( x ) , whereas probabilistic methods model a distribution ˆ p ( y | x ) to approximate the true p ( y | x ) .

4 TARFVAE

4.1 Overview

The architecture of TARFVAE, as illustrated in Figure 1 , builds upon a conditional V AE where input history x conditions the generation of target series y . The Gaussian posterior assumption in Section

3.1 fundamentally limits reconstruction accuracy in conventional VAEs, as Gaussian distributions constitute a tiny subset of all possible posterior distributions. To overcome this expressiveness limitation, we introduce TARFLOW to refine an initial Gaussian posterior q ( z 0 | x, y ) into a flexible complex distribution that closely matches the true posterior. Specifically, both x and y are utilized for training, whereas only x is used for inference.

Instance normalization . Following many state-of-the-art models[4, 5, 17, 39], we first apply instance normalization[40] to remove the local statistics of the input history to stabilize the base prediction, and then restore them to the model prediction. Notably, during training, both x and y are normalized based on x 's mean and variance without using y 's statistics. This ensures x 's normalization remains consistent between training and inference, regardless of y 's presence.

Series embedding . Then we perform series embedding[5, 17] on the needed time series for the encoder, decoder, prior and flow modules independently. Each module employs a dedicated embedding layer to extract module-specific temporal patterns. This architectural isolation prevents cross-module interference while enabling targeted feature learning.

One-step generation . The prior and encoder modules parameterize two Gaussians, the prior p ( z | x ) and initial posterior q ( z 0 | x, y ) , respectively. During training, the flow module transforms z 0 ∼ q ( z 0 | x, y ) to approximate z from the true posterior, while inference directly samples z ∼ p ( z | x ) . Finally, the decoder maps z to ˆ y conditioned on x , enabling efficient one-step full-horizon generation.

The training and inference process of TARFVAE can be formulated as follows:

Training

Inference

Below, we describe the TARFVAE implementation used in our work, while noting that alternative approaches could be adopted.

(1t1）exp（s

分

training only

inferenceonly

2

1

permutation

K2

2

$2

k-

Input

1

indino

Time

C

C

xK

sufficient y ~p(y|z,c)

Encoder

20 ~ q(z0/x,y)

TARFLOW

2~q(2/x,y)

Channel

y

c,y

..........................

Decoder

Prior

2~p(2|c)

Time

Channel

4.2 MLP Foundation

Although many Transformer-based architectures such as PatchTST[4], iTranformer[5] and DUET[39] have demonstrated that self-attention can be effective for long-term forecasting, recent studies[14, 15] argue that sophisticated designs might not be necessary and suggest simple models like feedforward neural networks could suffice for the job. To thoroughly test the TARFV AE framework's effectiveness and avoid confounding effects from elaborately designed temporal processing modules, we implement simple MLP blocks as shown in Figure 2 for basic input mixing, where the channel-wise mappings enable the capture of inter-channel dependencies. These MLP blocks form the architectural foundation across our encoder, decoder, and prior module.

intra-channel MLP

inter-channel MLP

RCxd

IRCxd

Dropout

LayerNorm

Transpose

FC

Dropout

Transpose

LayerNorm

In the prior module and encoder, following multiple MLP blocks, linear layers output the estimated mean and variance for the Gaussian prior p ( z | x ) and initial posterior q ( z 0 | x, y ) respectively as follows:

The decoder aims to reconstruct y using z conditioned on x . We chose to augment z via connecting a full attention output where z queries out relevant information from x . Subsequently, MLP blocks process this mixed information, and a linear projection is used to achieve the final reconstruction of y. The process is outlined as follows:

4.3 TARFLOW

In the flow module, the input z 0 and all intermediate latent variables { z k } ( k = 1 , 2 , ..., K ) maintain consistent dimensionality R C × D , where D represents the latent dimension. These variables are naturally partitioned by channels, and we perform the transformation (12) conditioned on x, y in each TARFLOW block as follows:

Following the original implementation[28], we adopt a dimension-reversing permutation between adjacent blocks. This module finally outputs z K ≈ z .

Combining (10), (11) and (6), the training loss for a single sample generation is derived as

where p ( y | z, x ) is chosen to be Guassian for continuous predictions and its mean µ y and variance σ 2 y are implemented as ˆ y and I in our work.

5 Experiments

5.1 Setup

Datasets . To comprehensively evaluate the performance of our proposed TARFVAE, we conduct extensive experiments on 8 widely-used real-world datasets: four ETT subsets (ETTh1, ETTh2, ETTm1, ETTm2), Electricity, Exchange, Weather[6, 7], and Solar-Energy[41].

Baselines . We extensively choose the recent state-of-the-art models to serve as baselines. For deterministic methods, we include Linear-based or MLP-based methods (SOFTS[17], TiDE[16], TSMixer[9] and DLinear[14]) and Transformer-based models (DUET[39], iTransformer[5], PatchTST[4], Crossformer[30] and FEDformer[8]). For generative approaches, given their distinct evaluation protocols compared to deterministic approaches, we follow mr-Diff[36] and compare against both mr-Diff and its benchmarked generative baselines: TimeDiff[13], TimeGrad[10], CSDI[11], SSSD[35], D 3 VAE[22], CPF[42] and PSA-GAN[43].

Implementation details . When comparing with deterministic models, the long-term forecasting benchmarks follow the common setting[5-7, 17], with the lookback window length L set to 96 and the prediction horizon H to { 96 , 192 , 336 , 720 } for all datasets. For comparison with mr-Diff and its benchmarked baselines, we adopt the same configurations: H is 168 for Electricity and ETTh1, 192 for ETTm1, and 672 for Weather, while L is chosen from { 96 , 192 , 336 , 720 , 1440 } . Mean Squared Error (MSE) and Mean Absolute Error (MAE) are adopted as evaluation metrics. Since we can sample different sizes of results once our model is trained, we calculate MSE and MAE for the median of sampled results. We also compute the Continuous Ranked Probability Score (CRPS)[44] based on sampled results as a probabilistic forecasting metric. All our experiments are implemented using PyTorch[45] on a single Nvidia-H20 GPU with 141 GB memory, except for the inference efficiency comparison experiment which is conducted on an Nvidia-A6000 GPU with 48 GB memory to align with the experimental settings of the compared generative baselines. Our training process is guided by the loss function (28) and employs the ADAM optimizer, and the best model is selected based on the MSE of the median of 50 generated samples on the validation set.

5.2 Main results

Table 1 presents the multivariate time series forecasting performance of TARFVAE in contrast with selected deterministic baselines, where the results of sampling 200 are shown and the full results of different sample sizes can be seen in Table 2 . The results demonstrate that TARFVAE achieves the highest number of best outcomes across various forecast horizons in all 8 datasets. Notably, TARFVAE consistently achieves superior performance under varing forecast horizons in 5 out of 8 datasets with significant improvements. For instance, it reduces average MSE by 4 . 3% on ETTh2 and 4 . 8% on Weather, respectively. These results not only demonstrate TARFVAE's precision advantages compared to state-of-the-art deterministic methods but also reveal its ability to produce robust performance in considerable long-term forecasting (e.g., 720-horizon predictions).

Table 3 shows the performance of TARFVAE compared to other generative methods. TARFVAE achieves the best overall performance, securing top-1 positions in 5 out of 10 metrics and ranking second in 2 metrics. To compare inference efficiency, we rerun TARFVAE under the same configurations as the baselines, since some baselines such as mr-Diff and TimeDiff are not open-sourced. The results shown in Table 4 demonstrate TARFVAE's significant speed advantages: for instance, it reduces inference time by 85 . 6% compared to mr-Diff at horizon H = 720 in the ETTh1 dataset, and even more so when compared to other methods. Furthermore, TARFVAE maintains computational stability when handling extended forecasting horizons - only the final linear layer's dimensionality increases, while other generative methods require progressive time increments. Additionally, its parallelized sampling computation ensures minimal latency variation (from 1 to 200 samples) under moderate sampling scales. These findings collectively establish TARFVAE as both an accurate and efficient approach, enabling high-quality probabilistic forecasts through one-step generation while outperforming state-of-the-art generative methods in computational scalability.

Models

TARFVAE

DUET

SOFTS

iTransformer

TSMixer

PatchTST

Crossformer

TiDE

DLinear

FEDformer

(ours)

(2025)

(2024)

(2024)

(2023)

(2023)

(2023)

(2023)

(2023)

(2022)

Metric

MSE

MAE

MSE

MAE

MSE

MAE

MSE

MAE

MSE

MAE

MSE

MAE

MSE

MAE

MSE

MAE

MSE

MAE

MSE

MAE

ETTh1

96

0.361

0.388

0.377

0.393

0.381

0.399

0.386

0.405

0.401

0.412

0.394

0.406

0.423

0.448

0.479

0.464

0.386

0.400

0.376

0.419

192

0.410

0.422

0.429

0.425

0.435

0.431

0.441

0.436

0.452

0.442

0.440

0.435

0.471

0.474

0.525

0.492

0.437

0.432

0.420

0.448

336

0.455

0.445

0.471

0.446

0.480

0.452

0.487

0.458

0.492

0.463

0.491

0.462

0.570

0.546

0.565

0.515

0.481

0.459

0.459

0.465

720

0.481

0.469

0.496

0.480

0.499

0.488

0.503

0.491

0.507

0.490

0.487

0.479

0.653

0.621

0.594

0.558

0.519

0.516

0.506

0.507

Avg

0.427

0.431

0.443

0.436

0.449

0.443

0.454

0.448

0.463

0.452

0.453

0.446

0.529

0.522

0.541

0.507

0.456

0.452

0.440

0.459

ETTh2

96

0.273

0.329

0.296

0.345

0.297

0.347

0.297

0.349

0.319

0.361

0.288

0.340

0.745

0.584

0.400

0.440

0.333

0.387

0.358

0.397

192

0.359

0.382

0.368

0.389

0.373

0.394

0.380

0.400

0.402

0.410

0.376

0.395

0.877

0.656

0.528

0.509

0.477

0.476

0.429

0.439

336

0.391

0.409

0.411

0.422

0.410

0.426

0.428

0.432

0.444

0.446

0.440

0.451

1.043

0.731

0.643

0.571

0.594

0.541

0.496

0.487

720

0.399

0.428

0.412

0.434

0.411

0.433

0.427

0.445

0.441

0.450

0.436

0.453

1.104

0.763

0.874

0.679

0.831

0.657

0.463

0.474

Avg

0.355

0.387

0.372

0.398

0.373

0.400

0.383

0.407

0.402

0.417

0.385

0.410

0.942

0.684

0.611

0.550

0.559

0.515

0.437

0.449

ETTm1

96

0.311

0.351

0.324

0.354

0.325

0.361

0.334

0.368

0.323

0.363

0.329

0.365

0.404

0.426

0.364

0.387

0.345

0.372

0.379

0.419

192

0.361

0.378

0.369

0.379

0.375

0.389

0.377

0.391

0.376

0.392

0.380

0.394

0.450

0.451

0.398

0.404

0.380

0.389

0.426

0.441

336

0.391

0.401

0.404

0.402

0.405

0.412

0.426

0.420

0.407

0.413

0.400

0.410

0.532

0.515

0.428

0.425

0.413

0.413

0.445

0.459

720

0.456

0.435

0.463

0.437

0.466

0.447

0.491

0.459

0.485

0.459

0.475

0.453

0.666

0.589

0.487

0.461

0.474

0.453

0.543

0.490

Avg

0.380

0.391

0.390

0.393

0.393

0.402

0.407

0.410

0.398

0.407

0.396

0.406

0.513

0.495

0.419

0.419

0.403

0.407

0.448

0.452

ETTm2

96

0.171

0.249

0.174

0.255

0.180

0.261

0.180

0.264

0.182

0.266

0.184

0.264

0.287

0.366

0.207

0.305

0.193

0.292

0.203

0.287

192

0.229

0.291

0.243

0.302

0.246

0.306

0.250

0.309

0.249

0.309

0.246

0.306

0.414

0.492

0.290

0.364

0.284

0.362

0.269

0.328

336

0.293

0.334

0.304

0.341

0.319

0.352

0.311

0.348

0.309

0.347

0.308

0.346

0.597

0.542

0.377

0.422

0.369

0.427

0.325

0.366

720

0.391

0.391

0.399

0.397

0.405

0.401

0.412

0.407

0.416

0.408

0.409

0.402

1.730

1.042

0.558

0.524

0.554

0.522

0.421

0.415

Avg

0.271

0.316

0.280

0.324

0.288

0.330

0.288

0.332

0.289

0.333

0.287

0.330

0.757

0.611

0.358

0.404

0.350

0.401

0.305

0.349

Exchange

96

0.086

0.204

0.086

0.205

0.090

0.211

0.086

0.206

0.166

0.316

0.088

0.205

0.256

0.367

0.094

0.218

0.088

0.218

0.148

0.278

192

0.172

0.294

0.182

0.305

0.182

0.304

0.177

0.299

0.279

0.402

0.176

0.299

0.470

0.509

0.184

0.307

0.176

0.315

0.271

0.315

336

0.340

0.423

0.310

0.403

0.363

0.438

0.331

0.417

0.477

0.548

0.301

0.397

1.268

0.883

0.349

0.431

0.313

0.427

0.460

0.427

720

0.797

0.678

0.693

0.624

0.997

0.743

0.847

0.691

0.654

0.662

0.901

0.714

1.767

1.068

0.852

0.698

0.839

0.695

1.195

0.695

Avg

0.349

0.400

0.318

0.384

0.408

0.424

0.360

0.403

0.394

0.482

0.367

0.404

0.940

0.707

0.370

0.414

0.354

0.414

0.519

0.429

Electricity

96

0.139

0.238

0.145

0.233

0.143

0.233

0.148

0.240

0.157

0.260

0.164

0.251

0.219

0.314

0.237

0.329

0.197

0.282

0.193

0.308

192

0.159

0.255

0.163

0.248

0.158

0.248

0.162

0.253

0.173

0.274

0.173

0.262

0.231

0.322

0.236

0.330

0.196

0.285

0.201

0.315

336

0.173

0.270

0.175

0.262

0.178

0.269

0.178

0.269

0.192

0.295

0.190

0.279

0.246

0.337

0.249

0.344

0.209

0.301

0.214

0.329

720

0.202

0.294

0.204

0.291

0.218

0.305

0.225

0.317

0.223

0.318

0.230

0.313

0.280

0.363

0.284

0.373

0.245

0.333

0.246

0.355

Avg

0.168

0.264

0.172

0.259

0.174

0.264

0.178

0.270

0.186

0.287

0.189

0.276

0.244

0.334

0.252

0.344

0.212

0.300

0.214

0.327

Solar

96

0.195

0.220

0.200

0.207

0.200

0.230

0.203

0.237

0.221

0.275

0.205

0.246

0.310

0.331

0.312

0.399

0.290

0.378

0.242

0.342

192

0.225

0.242

0.228

0.233

0.229

0.253

0.233

0.261

0.268

0.306

0.237

0.267

0.734

0.725

0.339

0.416

0.320

0.398

0.285

0.380

336

0.250

0.262

0.262

0.244

0.243

0.269

0.248

0.273

0.272

0.294

0.250

0.276

0.750

0.735

0.368

0.430

0.353

0.415

0.282

0.376

720

0.254

0.269

0.258

0.249

0.245

0.272

0.249

0.275

0.281

0.313

0.252

0.275

0.769

0.765

0.370

0.425

0.356

0.413

0.357

0.427

Avg

0.231

0.248

0.237

0.233

0.229

0.256

0.233

0.262

0.261

0.297

0.236

0.266

0.641

0.639

0.347

0.418

0.330

0.401

0.292

0.381

Weather

96

0.151

0.197

0.163

0.202

0.166

0.208

0.174

0.214

0.166

0.210

0.176

0.217

0.158

0.230

0.202

0.261

0.196

0.255

0.217

0.296

192

0.202

0.242

0.218

0.252

0.217

0.253

0.221

0.254

0.215

0.256

0.221

0.256

0.206

0.277

0.242

0.298

0.237

0.296

0.276

0.336

336

0.262

0.289

0.274

0.294

0.282

0.300

0.278

0.296

0.287

0.300

0.275

0.296

0.272

0.335

0.287

0.335

0.283

0.335

0.339

0.380

720

0.342

0.341

0.349

0.343

0.356

0.351

0.358

0.347

0.355

0.348

0.352

0.346

0.398

0.418

0.351

0.386

0.345

0.381

0.403

0.428

Avg

0.239

0.267

0.251

0.273

0.255

0.278

0.258

0.278

0.256

0.279

0.256

0.279

0.259

0.315

0.271

0.320

0.265

0.317

0.309

0.360

1

st

Count

33

27

2

12

4

2

1

0

1

0

1

1

0

0

0

0

0

0

0

0

Datasets

ETTh1

ETTh2

ETTm1

ETTm2

Exchange

Electricity

Solar

Weather

Metric

MSE

MAE

CRPS

MSE

MAE

CRPS

MSE

MAE

CRPS

MSE

MAE

CRPS

MSE

MAE

CRPS

MSE

MAE

CRPS

MSE

MAE

CRPS

MSE

MAE

CRPS

96

20

0.376

0.399

0.323

0.275

0.331

0.302

0.321

0.359

0.296

0.177

0.257

0.215

0.087

0.206

0.176

0.143

0.243

0.203

0.201

0.224

0.183

0.154

0.200

0.173

50

0.366

0.391

0.315

0.274

0.330

0.298

0.314

0.354

0.289

0.173

0.252

0.210

0.086

0.204

0.173

0.141

0.240

0.198

0.197

0.221

0.179

0.152

0.198

0.170

100

0.363

0.389

0.313

0.273

0.329

0.297

0.312

0.352

0.287

0.171

0.250

0.209

0.086

0.204

0.172

0.140

0.239

0.197

0.196

0.220

0.177

0.151

0.198

0.169

200

0.361

0.388

0.311

0.273

0.329

0.296

0.311

0.351

0.286

0.171

0.249

0.208

0.086

0.204

0.171

0.139

0.238

0.196

0.195

0.220

0.177

0.151

0.197

0.169

192

20

0.423

0.430

0.355

0.361

0.384

0.355

0.373

0.387

0.321

0.233

0.296

0.254

0.176

0.298

0.254

0.163

0.260

0.219

0.230

0.245

0.207

0.207

0.246

0.210

50

0.415

0.425

0.347

0.359

0.382

0.351

0.365

0.381

0.313

0.230

0.293

0.249

0.173

0.295

0.248

0.160

0.257

0.215

0.227

0.243

0.202

0.204

0.243

0.206

100

0.412

0.423

0.344

0.359

0.382

0.350

0.363

0.379

0.311

0.229

0.292

0.247

0.172

0.294

0.246

0.159

0.256

0.213

0.226

0.243

0.201

0.203

0.242

0.204

200

0.410

0.422

0.342

0.359

0.382

0.349

0.361

0.378

0.309

0.229

0.291

0.247

0.172

0.294

0.245

0.159

0.255

0.212

0.225

0.242

0.200

0.202

0.242

0.203

336

20

0.465

0.452

0.385

0.393

0.412

0.380

0.404

0.411

0.340

0.301

0.339

0.293

0.338

0.422

0.381

0.179

0.276

0.230

0.254

0.266

0.225

0.269

0.293

0.252

50

0.458

0.447

0.378

0.391

0.410

0.375

0.395

0.404

0.332

0.295

0.335

0.288

0.339

0.422

0.376

0.175

0.272

0.224

0.252

0.263

0.220

0.264

0.290

0.246

100

0.456

0.446

0.375

0.391

0.410

0.373

0.392

0.402

0.329

0.294

0.334

0.286

0.340

0.423

0.374

0.174

0.271

0.223

0.251

0.262

0.219

0.263

0.289

0.244

200

0.455

0.445

0.374

0.391

0.409

0.372

0.391

0.401

0.328

0.293

0.334

0.285

0.340

0.423

0.373

0.173

0.270

0.222

0.250

0.262

0.218

0.262

0.289

0.244

720

20

0.502

0.483

0.414

0.403

0.431

0.394

0.470

0.445

0.373

0.400

0.397

0.345

0.792

0.676

0.647

0.208

0.299

0.253

0.258

0.272

0.234

0.349

0.345

0.300

50

0.488

0.473

0.406

0.400

0.429

0.389

0.461

0.439

0.365

0.394

0.393

0.339

0.796

0.678

0.640

0.204

0.295

0.247

0.256

0.270

0.230

0.344

0.342

0.295

100

0.483

0.471

0.403

0.399

0.428

0.388

0.458

0.437

0.362

0.393

0.391

0.336

0.796

0.678

0.637

0.203

0.294

0.246

0.255

0.269

0.228

0.343

0.341

0.293

200

0.481

0.469

0.401

0.399

0.428

0.387

0.456

0.435

0.360

0.391

0.391

0.335

0.797

0.678

0.636

0.202

0.294

0.245

0.254

0.269

0.227

0.342

0.341

0.292

Avg

20

0.442

0.441

0.369

0.358

0.390

0.358

0.392

0.400

0.332

0.278

0.322

0.277

0.348

0.400

0.365

0.173

0.269

0.226

0.236

0.252

0.212

0.245

0.271

0.234

50

0.432

0.434

0.361

0.356

0.388

0.353

0.384

0.394

0.325

0.273

0.318

0.271

0.349

0.400

0.359

0.170

0.266

0.221

0.233

0.249

0.208

0.241

0.268

0.229

100

0.428

0.432

0.359

0.356

0.387

0.352

0.381

0.392

0.322

0.272

0.317

0.270

0.349

0.400

0.357

0.169

0.265

0.219

0.232

0.249

0.206

0.240

0.267

0.228

200

0.427

0.431

0.357

0.355

0.387

0.351

0.380

0.391

0.321

0.271

0.316

0.269

0.349

0.400

0.356

0.168

0.264

0.219

0.231

0.248

0.206

0.239

0.267

0.227

To ensure a systematic comparison of CRPS performance, we further benchmark our model against the baselines in ProbTS[27] under an identical CRPS protocol. The results in Table 5 show our TARFVAE still outperforms all probabilistic and point competitors in CRPS except for ranking 2nd in one case, demonstrating the effectiveness of our generative framework. To the best of our knowledge, this work represents the first demonstration of a generative model achieving comprehensive longterm forecasting across diverse datasets (with the number of channels ranging from 7 in ETT to 321 in Electricity) while simultaneously providing state-of-the-art deterministic (MSE, MAE) and probabilistic (CRPS) evaluation metrics.

5.3 Ablation study

In this part, we conduct experiments using the exact same configuration as the main experiments, modifying only the targeted settings for analysis to ensure the consistency and reliability of the conclusions.

Influence of lookback windows . Figure 3 demonstrates the performance of TARFVAE with varying lookback window lengths. Overall, TARFVAE consistently achieves superior performance across all evaluated lookback windows, with its accuracy improving as the length gets longer. Notably, on the

ETTm2 dataset, TARFVAE exhibits sustained performance enhancement with increasing window lengths, while competing methods show clear saturation effects.

ETTm1

ETTm2

Weather

0.55

PatchTST

PatchTST

PatchTST

iTransformer

0.30

iTransformer

0.28

iTransformer

0.50

SOFTS

SOFTS

SOFTS

TarfVAE

TarfVAE

TarfVAE

MSE

0.45

MSI

汇

0.28

MSI

0.26

0.40

0.24

0.26

0.35

48

96

192

336

48

96

192

336

48

96

192

336

LookbackWindowLength

LookbackWindowLength

LookbackWindowLength

Architecture ablations . As evidenced in Table 6 , removing the TARFLOW component and further eliminating VAE lead to a sustained deterioration. These findings collectively demonstrate that the synergistic integration of TARFLOW with VAE critically enables TARFVAE's superior modeling capacity.

Influence of inference sample size . Upon completing model training, we evaluate TARFVAE's inference performance under varying sample sizes, as shown in Table 2 . As the sample size increases from 20 to 200, the reduced sampling bias consistently improves the MSE, MAE and CRPS. The absence of significant metric fluctuations across different sample sizes confirms the stability of TARFVAE's performance. These collective findings substantiate that TARFVAE achieves both reliable uncertainty quantification and deterministic modeling with strong operational stability.

6 Conclusion

In this work we present TARFVAE, an efficient one-step generative framework for time series forecasting. It integrates TARFLOW with VAE architecture to enhance posterior estimation, enabling the learning of a more expressive latent space which can lead to efficient modeling of the targeted time series distribution. Implemented on an simple MLP foundation in our extensive experiments, TARFVAE achieves superior or comparable performance to recent state-of-the-art deterministic and generative models across widely-used real-world benchmarks. To our knowledge, this is the first time a generative time series forecasting model delivers comprehensive long-term predictions consistent with deterministic methods on rich datasets, encompassing both deterministic (MSE, MAE) and probabilistic (CRPS) forecasting metrics. Meanwhile, TARFVAE's one-step generation reduces inference time by over 85% compared to other state-of-the-art generative models in long-term forecasting. Other implementations could be tried to achieve better performance in the future.

References

Weiwei Fang, Wenhao Zhuo, Jingwen Yan, Youyi Song, Dazhi Jiang, and Teng Zhou. Attention meets long short-term memory: A deep learning network for traffic flow forecasting. Physica A: Statistical Mechanics and its Applications, 587:126485, 2022.

Mohammad Amin Morid, Olivia R Liu Sheng, and Joseph Dunbar. Time series prediction using deep learning methods in healthcare. ACM Transactions on Management Information Systems, 14(1):1-29, 2023.

Meng Qi, Yuanyuan Shi, Yongzhi Qi, Chenxin Ma, Rong Yuan, Di Wu, and Zuo-Jun Shen. A practical end-to-end inventory management model with deep learning. Management Science, 69(2):759-773, 2023.

Yuqi Nie, Nam H. Nguyen, Phanwadee Sinthong, and Jayant Kalagnanam. A time series is worth 64 words: Long-term forecasting with transformers. In International Conference on Learning Representations, 2023.

Yong Liu, Tengge Hu, Haoran Zhang, Haixu Wu, Shiyu Wang, Lintao Ma, and Mingsheng Long. itransformer: Inverted transformers are effective for time series forecasting. In The Twelfth International Conference on Learning Representations, 2023.

Haoyi Zhou, Shanghang Zhang, Jieqi Peng, Shuai Zhang, Jianxin Li, Hui Xiong, and Wancai Zhang. Informer: Beyond efficient transformer for long sequence time-series forecasting. In Proceedings of the AAAI conference on artificial intelligence, volume 35, pages 11106-11115, 2021.

Haixu Wu, Jiehui Xu, Jianmin Wang, and Mingsheng Long. Autoformer: Decomposition transformers with auto-correlation for long-term series forecasting. Advances in neural information processing systems, 34:22419-22430, 2021.

Tian Zhou, Ziqing Ma, Qingsong Wen, Xue Wang, Liang Sun, and Rong Jin. Fedformer: Frequency enhanced decomposed transformer for long-term series forecasting. In International conference on machine learning, pages 27268-27286. PMLR, 2022.

Vijay Ekambaram, Arindam Jati, Nam Nguyen, Phanwadee Sinthong, and Jayant Kalagnanam. Tsmixer: Lightweight mlp-mixer model for multivariate time series forecasting. In Proceedings of the 29th ACM SIGKDD conference on knowledge discovery and data mining, pages 459469, 2023.

Kashif Rasul, Calvin Seward, Ingmar Schuster, and Roland Vollgraf. Autoregressive denoising diffusion models for multivariate probabilistic time series forecasting. In International conference on machine learning, pages 8857-8868. PMLR, 2021.

Yusuke Tashiro, Jiaming Song, Yang Song, and Stefano Ermon. Csdi: Conditional score-based diffusion models for probabilistic time series imputation. Advances in neural information processing systems, 34:24804-24816, 2021.

David Salinas, Valentin Flunkert, Jan Gasthaus, and Tim Januschowski. Deepar: Probabilistic forecasting with autoregressive recurrent networks. International journal of forecasting, 36(3): 1181-1191, 2020.

Lifeng Shen and James Kwok. Non-autoregressive conditional diffusion models for time series prediction. In International Conference on Machine Learning, pages 31016-31029. PMLR, 2023.

Ailing Zeng, Muxi Chen, Lei Zhang, and Qiang Xu. Are transformers effective for time series forecasting? In Proceedings of the AAAI conference on artificial intelligence, volume 37, pages 11121-11128, 2023.

Fan-Keng Sun, Yu-Cheng Wu, and Duane S Boning. Simple feedfoward neural networks are almost all you need for time series forecasting. arXiv preprint arXiv:2503.23621, 2025.

Abhimanyu Das, Weihao Kong, Andrew Leach, Shaan K Mathur, Rajat Sen, and Rose Yu. Long-term forecasting with tiDE: Time-series dense encoder. Transactions on Machine Learning Research, 2023. ISSN 2835-8856. URL https://openreview.net/forum?id= pCbC3aQB5W .

Lu Han, Xu-Yang Chen, Han-Jia Ye, and De-Chuan Zhan. Softs: Efficient multivariate time series forecasting with series-core fusion. In The Thirty-eighth Annual Conference on Neural Information Processing Systems, 2024.

Abhyuday Desai, Cynthia Freeman, Zuhui Wang, and Ian Beaver. Timevae: A variational auto-encoder for multivariate time series generation. arXiv preprint arXiv:2111.08095, 2021.

Prafulla Dhariwal and Alexander Nichol. Diffusion models beat gans on image synthesis. Advances in neural information processing systems, 34:8780-8794, 2021.

William Harvey, Saeid Naderiparizi, Vaden Masrani, Christian Weilbach, and Frank Wood. Flexible diffusion modeling of long videos. Advances in Neural Information Processing Systems, 35:27953-27965, 2022.

Chitwan Saharia, William Chan, Saurabh Saxena, Lala Li, Jay Whang, Emily L Denton, Kamyar Ghasemipour, Raphael Gontijo Lopes, Burcu Karagol Ayan, Tim Salimans, et al. Photorealistic text-to-image diffusion models with deep language understanding. Advances in neural information processing systems, 35:36479-36494, 2022.

Yan Li, Xinjiang Lu, Yaqing Wang, and Dejing Dou. Generative time series forecasting with diffusion, denoise, and disentanglement. Advances in Neural Information Processing Systems, 35:23009-23022, 2022.

Diederik P Kingma, Max Welling, et al. Auto-encoding variational bayes, 2013.

Borui Cai, Shuiqiao Yang, Longxiang Gao, and Yong Xiang. Hybrid variational autoencoder for time series forecasting. Knowledge-Based Systems, 281:111079, 2023.

Wenchao Chen, Long Tian, Bo Chen, Liang Dai, Zhibin Duan, and Mingyuan Zhou. Deep variational graph convolutional recurrent network for multivariate time series anomaly detection. In International conference on machine learning, pages 3621-3633. PMLR, 2022.

Zhiyuan Wang, Xovee Xu, Weifeng Zhang, Goce Trajcevski, Ting Zhong, and Fan Zhou. Learning latent seasonal-trend representations for time series forecasting. Advances in Neural Information Processing Systems, 35:38775-38787, 2022.

Jiawen Zhang, Xumeng Wen, Zhenwei Zhang, Shun Zheng, Jia Li, and Jiang Bian. Probts: Benchmarking point and distributional forecasting across diverse prediction horizons. Advances in Neural Information Processing Systems, 37:48045-48082, 2024.

Shuangfei Zhai, Ruixiang Zhang, Preetum Nakkiran, David Berthelot, Jiatao Gu, Huangjie Zheng, Tianrong Chen, Miguel Angel Bautista, Navdeep Jaitly, and Josh Susskind. Normalizing flows are capable generative models. arXiv preprint arXiv:2412.06329, 2024.

Xiangfei Qiu, Hanyin Cheng, Xingjian Wu, Jilin Hu, Chenjuan Guo, and Bin Yang. A comprehensive survey of deep learning for multivariate time series forecasting: A channel strategy perspective. arXiv preprint arXiv:2502.10721, 2025.

Yunhao Zhang and Junchi Yan. Crossformer: Transformer utilizing cross-dimension dependency for multivariate time series forecasting. In The Eleventh International Conference on Learning Representations, 2023. URL https://openreview.net/forum?id=vSVLM2j9eie .

Lu Han, Han-Jia Ye, and De-Chuan Zhan. The capacity and robustness trade-off: Revisiting the channel independent strategy for multivariate time series forecasting. IEEE Transactions on Knowledge and Data Engineering, 2024.

Kashif Rasul, Abdul-Saboor Sheikh, Ingmar Schuster, Urs Bergmann, and Roland Vollgraf. Multivariate Probabilistic Time Series Forecasting via Conditioned Normalizing Flows. In International Conference on Learning Representations 2021, 2021. URL https: //openreview.net/forum?id=WiGQBFuVRv .

George Papamakarios, Eric Nalisnick, Danilo Jimenez Rezende, Shakir Mohamed, and Balaji Lakshminarayanan. Normalizing flows for probabilistic modeling and inference. Journal of Machine Learning Research, 22(57):1-64, 2021.

George Papamakarios, Theo Pavlakou, and Iain Murray. Masked autoregressive flow for density estimation. Advances in neural information processing systems, 30, 2017.

Juan Miguel Lopez Alcaraz and Nils Strodthoff. Diffusion-based time series imputation and forecasting with structured atate apace models. Transactions on machine learning research, pages 1-36, 2023.

Lifeng Shen, Weiyu Chen, and James Kwok. Multi-resolution diffusion models for time series forecasting. In The Twelfth International Conference on Learning Representations, 2024.

Esteban G Tabak and Cristina V Turner. A family of nonparametric density estimation algorithms. Communications on Pure and Applied Mathematics, 66(2):145-164, 2013.

Laurent Dinh, Jascha Sohl-Dickstein, and Samy Bengio. Density estimation using real nvp. arXiv preprint arXiv:1605.08803, 2016.

Xiangfei Qiu, Xingjian Wu, Yan Lin, Chenjuan Guo, Jilin Hu, and Bin Yang. Duet: Dual clustering enhanced multivariate time series forecasting. arXiv preprint arXiv:2412.10859, 2024.

Taesung Kim, Jinhee Kim, Yunwon Tae, Cheonbok Park, Jang-Ho Choi, and Jaegul Choo. Reversible instance normalization for accurate time-series forecasting against distribution shift. In International conference on learning representations, 2021.

Guokun Lai, Wei-Cheng Chang, Yiming Yang, and Hanxiao Liu. Modeling long-and short-term temporal patterns with deep neural networks. In The 41st international ACM SIGIR conference on research & development in information retrieval, pages 95-104, 2018.

Syama Sundar Rangapuram, Shubham Kapoor, Rajbir Singh Nirwan, Pedro Mercado, Tim Januschowski, Yuyang Wang, and Michael Bohlke-Schneider. Coherent probabilistic forecasting of temporal hierarchies. In International Conference on Artificial Intelligence and Statistics, pages 9362-9376. PMLR, 2023.

Paul Jeha, Michael Bohlke-Schneider, Pedro Mercado, Shubham Kapoor, Rajbir Singh Nirwan, Valentin Flunkert, Jan Gasthaus, and Tim Januschowski. Psa-gan: Progressive self attention gans for synthetic time series. In The Tenth International Conference on Learning Representations, 2022.

James E Matheson and Robert L Winkler. Scoring rules for continuous probability distributions. Management science, 22(10):1087-1096, 1976.

Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, et al. Pytorch: An imperative style, high-performance deep learning library. Advances in Neural Information Processing Systems, 32, 2019.

NeurIPS Paper Checklist

1. Claims

Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?

Answer: [Yes]

Justification: The abstract and introduction clearly help readers to understand our contributions and scope of TARFVAE, a novel model combining TARFlow and conditional VAE designed for long-term forecasting and uncertainty quantification.

Guidelines:

The answer NA means that the abstract and introduction do not include the claims made in the paper.

The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.

The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.

It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper.

2. Limitations

Question: Does the paper discuss the limitations of the work performed by the authors?

Answer: [Yes]

Justification: We have discussed the limitations of our work in Appendix B.

Guidelines:

The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.

The authors are encouraged to create a separate "Limitations" section in their paper.

The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.

The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.

The authors should reflect on the factors that influence the performance of the approach. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.

The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.

If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness.

While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.

3. Theory assumptions and proofs

Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?

Answer: [Yes]

Justification: We provide numerous formulas to support the complete derivation of the theory in Section 3 and Section 4.3.

Guidelines:

The answer NA means that the paper does not include theoretical results.

All the theorems, formulas, and proofs in the paper should be numbered and crossreferenced.

All assumptions should be clearly stated or referenced in the statement of any theorems.

The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition.

Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material.

Theorems and Lemmas that the proof relies upon should be properly referenced.

4. Experimental result reproducibility

Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?

Answer: [Yes]

Justification: We provide full information needed to reproduce the main experimental results in Section 5.1.

Guidelines:

The answer NA means that the paper does not include experiments.

If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.

If the contribution is a dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable.

Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model. In general. releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g., in the case of a large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed.

While NeurIPS does not require releasing code, the conference does require all submissions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example

(a) If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm.

(b) If the contribution is primarily a new model architecture, the paper should describe the architecture clearly and fully.

(c) If the contribution is a new model (e.g., a large language model), then there should either be a way to access this model for reproducing the results or a way to reproduce the model (e.g., with an open-source dataset or instructions for how to construct the dataset).

(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results.

5. Open access to data and code

Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?

Answer: [Yes]

Justification: All experiments are based on open-source data, and our reproducible code will be made available through an anonymous GitHub repository. The code will be made public once the paper is accepted.

Guidelines:

The answer NA means that paper does not include experiments requiring code.

Please see the NeurIPS code and data submission guidelines ( https://nips.cc/ public/guides/CodeSubmissionPolicy ) for more details.

While we encourage the release of code and data, we understand that this might not be possible, so 'No' is an acceptable answer. Papers cannot be rejected simply for not including code, unless this is central to the contribution (e.g., for a new open-source benchmark).

The instructions should contain the exact command and environment needed to run to reproduce the results. See the NeurIPS code and data submission guidelines ( https: //nips.cc/public/guides/CodeSubmissionPolicy ) for more details.

The authors should provide instructions on data access and preparation, including how to access the raw data, preprocessed data, intermediate data, and generated data, etc.

The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines. If only a subset of experiments are reproducible, they should state which ones are omitted from the script and why.

At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable).

Providing as much information as possible in supplemental material (appended to the paper) is recommended, but including URLs to data and code is permitted.

6. Experimental setting/details

Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?

Answer: [Yes]

Justification: We provide a detailed description of the experimental setup in Section 5.1.

Guidelines:

The answer NA means that the paper does not include experiments.

The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them.

The full details can be provided either with the code, in appendix, or as supplemental material.

7. Experiment statistical significance

Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?

Answer: [Yes]

Justification: In section 5.3, we conduct ablation study using the exact same configuration as the main experiments, modifying only the targeted settings for analysis to ensure the consistency and reliability of the conclusions.

Guidelines:

The answer NA means that the paper does not include experiments.

The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper.

The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions).

The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.)

The assumptions made should be given (e.g., Normally distributed errors).

It should be clear whether the error bar is the standard deviation or the standard error of the mean.

It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a 96% CI, if the hypothesis of Normality of errors is not verified.

For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates).

If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.

8. Experiments compute resources

Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?

Answer: [Yes]

Justification: Section 5.1 specifies the runtime environment and computational resources employed.

Guidelines:

The answer NA means that the paper does not include experiments.

The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.

The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute.

The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper).

9. Code of ethics

Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines ?

Answer: [Yes]

Justification: We conform with the NeurIPS Code of Ethics in every respect.

Guidelines:

The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics.

If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics.

The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction).

10. Broader impacts

Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?

Answer: [Yes]

Justification: We have discussed both potential positive societal impacts and negative societal impacts of our work in Appendix C.

Guidelines:

The answer NA means that there is no societal impact of the work performed.

If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact.

Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.

The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster.

The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.

If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML).

11. Safeguards

Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?

Answer: [NA]

Justification: The paper poses no such risks.

Guidelines:

The answer NA means that the paper poses no such risks.

Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.

Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.

We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort.

12. Licenses for existing assets

Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?

Answer: [Yes]

Justification: We cite the original paper for all the baselines properly.

Guidelines:

The answer NA means that the paper does not use existing assets.

The authors should cite the original paper that produced the code package or dataset.

The authors should state which version of the asset is used and, if possible, include a URL.

The name of the license (e.g., CC-BY 4.0) should be included for each asset.

For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided.

If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset.

For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided.

If this information is not available online, the authors are encouraged to reach out to the asset's creators.

13. New assets

Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?

Answer: [NA]

Justification: The paper does not release new assets.

Guidelines:

The answer NA means that the paper does not release new assets.

Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc.

The paper should discuss whether and how consent was obtained from people whose asset is used.

At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.

14. Crowdsourcing and research with human subjects

Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?

Answer: [NA]

Justification: The paper does not involve crowdsourcing nor research with human subjects. Guidelines:

The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.

Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper.

According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.

15. Institutional review board (IRB) approvals or equivalent for research with human subjects

Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?

Answer: [NA]

Justification: The paper does not involve crowdsourcing nor research with human subjects.

Guidelines:

The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.

Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.

We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution.

For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.

16. Declaration of LLM usage

Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.

Answer: [NA]

Justification: The paper does not involve LLMs as any important, original, or non-standard components.

Guidelines:

The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components.

Please refer to our LLM policy ( https://neurips.cc/Conferences/2025/LLM ) for what should or should not be described.

A Calculation of CRPS

After the inference of sufficient samples, we employ an efficient quantile-based approach to calculate CRPS. This aims to mitigate computation costs in estimating CRPS with all samples. Given certain quantiles [ q 1 , q 2 , ..., q n ] , we first sort these samples and extract corresponding quantile values [ Q ( q 1 ) , Q ( q 2 ) , ..., Q ( q n )] , then compute CRPS using the following formulation:

We adopt a constant extrapolation, that is Q ( q ) = Q ( q 1 ) , ∀ q < q 1 and Q ( q ) = Q ( q n ) , ∀ q > q n . Thus, (29) can be further formulated as

As for (30), it equals to

As for (32), it equals to

As for (31), it is discretized and approximated as

The inplementation of this calculation process can be found in our code.

B Limitations and Future Works

We analyze the limitations of our work and briefly discuss several directions for future research. In our work, we employ identical MLP blocks for the encoder, decoder, and prior modules. This configuration might limit the forecasting performance of the proposed TARFVAE model. For instance, since the encoder, decoder, and prior typically perform different functions, we may flexibly design distinct architectures for these three components. Moreover, MLP blocks may not be sufficiently powerful to effectively extract time series patterns from the input data, potentially hindering TARFV AE's ability to learn complex time series patterns. In the future, exploring advanced temporal processing modules or deep learning models within the TARFVAE framework could be beneficial.

C Societal Impacts

Extensive experiments on real-world datasets demonstrate the TARFVAE's potential capability to significantly benefit various application fields, spanning from finance to energy, by providing accurate and reliable forecasts. However, there are potential negative societal impacts to consider. As a model that often interacts with sensitive and proprietary data, particularly in sectors like healthcare and finance, TARFVAE could inadvertently contribute to privacy risks if the data it processes is not adequately protected. Moreover, Organizations might also become overly reliant on automated forecasts, neglecting valuable insights from human experts, which can affect decision quality. To prevent the aforementioned negative societal impacts, data integrity and security strategies should be well-designed to ensure the deployment of TARFVAE enhances its beneficial societal effects while reducing possible adverse impacts.

Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes TARFVAE, a novel generative framework for time series forecasting that combines Transformer-based autoregressive flow (TARFLOW) with variational autoencoders (VAE). The key innovation is enabling efficient one-step generation for long-term forecasting by using only the forward process of TARFLOW to enhance VAE's posterior estimation, avoiding computationally expensive autoregressive inverse operations while breaking the Gaussian assumption for more informative latent representations.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\n1.  The paper addresses a genuine limitation in current generative time series forecasting - the computational inefficiency of recurrent operations and repeated denoising steps.\n2. The combination of normalizing flows with VAE for time series forecasting is creative. The specific innovation of using only the forward process of TARFLOW to enhance posterior estimation while avoiding inverse operations is technically sound and addresses the efficiency concerns. \n3. The one-step generation capability could be particularly valuable for real-world applications where prediction speed matters.\n4. The proposed method outperforms both deterministic and generative SOTA across multiple benchmarks.\n\nWeaknesses:\n\n1. The decision to integrate Transformer architecture specifically within the flow component appears contradictory to the paper's own argument that complex architectures might not be necessary for effective time series representation learning, and the authors should provide more compelling theoretical or empirical justification for this architectural choice.\n2. While the paper claims computational efficiency as a major advantage, it would be significantly better to provide comprehensive analysis of the actual time costs for the proposed method compared to baseline methods.\n3. The paper fails to adequately explain how the dimensionality of the latent space is determined and whether this dimensionality choice has any principled relationship to the forecast horizon length.\n\nQUESTIONS:\n1. Why choose MLP decoders over more expressive architectures for complex temporal patterns?\n2. How does the method scale with very long input sequences or forecast horizons?\n3. The claim that complex architectures aren't necessary needs stronger justification. Have you tested more complex alternatives?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-27"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proposes to treat time series prediction as conditional generation: given a lookback history, e.g. 96 consecutive observations, one may predict the next observations up to a prespecified horizon e.g. 192 observations. In such a setting, one may use the conditional VAE architecture where the historical observations are given as conditioning input, this defines a distribution in latent space from which we sample and decode to produce the full matrix of horizon predictions in one neural network pass. Sampling multiple latents and decoding yields multiple output predictions.\n\nThis is contrast to autoregressive methods that must compute predictions up to a horizon time step by time step each requiring a network pass, or diffusion that requires iterative denoising where each denoising step is a single network pass.\n\nAt a high level the method uses the standard C-VAE components, similar to a standard VAE however a conditioning  variable informs the prior over latent variable and the likelihood (decoder) and also the approximate posterior over the latent variable (encoder). Typically the approximate posterior is takes observed data as input and returns an isotrpoic Gaussian distribution, this work uses a Normalizing Flow model that incorporates a transformer, TARFLOW. During training, the past data and future data is passed to the encoder, the TARFLOW approximate posterior, the ELBO is used to inform the approximate posterior to align with the Gaussian prior (determined lookback data) and the likelihood of the output data (decoder reconstruction).\n\nAt inference time, the lookback data is used to inform the Gaussian prior over latent variables, a latent vector is sampled, and the lookback data as well as the sampled latent value are decoded to yield a matrix of predictions for the time series up to the horizon. The TARFLOW model is not required.\n\nExperiments are performed on a range of datasets from the literature and show favorable performance against a range of recent methods from the literature.\n\nSTRENGTHS AND WEAKNESSES:\n# Strengths\n- the approach seems simple, elegant, a Conditoinal VAE that uses modern architectures.\n\n# Weaknesses\n- fixed time length, auto regressive for longer time series than the model was trained for still requires multiple forward passes, can the model be used for settings where the lookback is shorter than whaat was trained? (Presumably the same limitation applies to diffusion based methods)\n- where is the baseline mrDIFF?\n- it seems the SOFTS and DUET authors report the mean of MSE and MAE of predictions, whereas this work shows the median. This makes the results not comparable (arguably median may be lower as it negates large MSE outliers and MSE>0 bounds low MSE outliers). If the mean MSE, MAE are still outperforming baselines, then I will happily raise my score.\n\n## Minor Comment\n- is it possible to see the MSE per time step? Does the MSE grow with time?\n- the hypothesis of the paper seems strange. In generative modeling, in the last few years, iterative refinement methods, using autoregression or diffusion, have been outperforming one-shot methods like VAEs and GANs. This paper argues that the Gaussian approximate posterior is to blame and simply making the approximate posterior more flexible during training gives it the advantage. I realise this is high level and speculative and the ablation study removing TARFLOW shows a decrement, but it may be worth acknowledging or discussing and make the paper more convincing: a more flexible approximate posterior can make VAEs outperform diffusion apparently.\n- the notation switch of learning $(x,y)$ in Sectoin 3 to $(x, y)$ in section 4 was rather confusing at first\n\nQUESTIONS:\n- how to handle test time sequences shorter than the lookback duration?\n- can the authors report mean MSE and MAE in line with the results copied from SOFTS and DUET?\n- can the authors elaborate on why a Conditional-VAE outperforms diffusion somewhat opposing the community consensus that diffusion\n\nLIMITATIONS:\n- input and outpout shapes are fixed during training (which is true for some diffusion methods too)", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe work proposes TARFVAE, an approach that integrates transformer-based auto-regressive flows (TARF) with VAE, for time series forecasting task. It uses TARFs to model variational posterior latent distributions which are more expressive than parameterized Gaussian distributions, which is typically used in VAE. The approach also permits one-step generation for future forecasting, achieving great computational efficiency. As a generative model, the proposed approach shows performance and efficiency superior to both deterministic and generative models on various time series datasets, particularly in long-term forecasting.\n\nSTRENGTHS AND WEAKNESSES:\n## Strengths:\n1. Experiment results on standard benchmark datasets outperformed the SOTA models while preserving generation capability and great computational efficiency due to one-step generation. The experiment results provides solid support to the claimed contributions of the work.\n2. The work provides rich technical details when describing the TARFVAE method. The presentation is easy to follow and read.\n\n## Weakness:\n1. My biggest concern of the work is the limited novelty despite its superior performance and computational efficiency. The approach of using normalizing flows to model the variational posterior distribution in VAE has been explored in previous works [1]. The work can be positioned as an work that extend this approach to the time-series domain.\n2. The major experiment results are based on an MLP architecture proposed in the work. Exploring other model architectures like transformers could both highlight the generalizability and flexibility of the work and makes the experiment results more solid.\n\nReferences:\n\n[1] Rezende, Danilo, and Shakir Mohamed. \"Variational inference with normalizing flows.\" ICML 2015\n\nQUESTIONS:\nPlease see the weakness section.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces a new generative model for time-series forecasting, where the key idea is to combine autoregressive flow with autoencoder to achieve efficient one-step generation.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\n\n- Overall, I think this is a good paper, introducing an **efficient** generative model for time-series forecasting.\n- Specifically, I like Section 4.2 MLP Foundation, which attempts to seperate the confounding effects of different network architectures as this paper mainly focuses on the generative forecasting framework. \n\nWeaknesses\n\n- My major concerns stem from the facts that although this paper proposes a generative forecasting framework, it **merely compares with recent point forecasting models** and **mainly consider point forecasting metrics**.\n  - Typically, point and probabilistic forecasting studies evolve in their respective research threads and rarely mix with each other's branch. Recently, https://arxiv.org/abs/2310.07446 provided a systematic study comparing these two research branches, and according to their analyses, one major disadvantage of existing probabilistic forecasting methods lies in their inefficiency because many of them mainly considered short-term forecasting.\n  - From the efficiency perspective, I think this paper has addressed a pain point of existing probabilistic models, but the experimental parts do not involve comparisons with previous probabilistic forecasting models.\n  - If this paper focuses on generative models, delivering probabilistic forecasts rather than point forecasts, why the main experimental comparisons (Table 1) are presented using point metrics, such as MSE and MAE? Only in Table 2, I have found the results using the CRPS metrics, while only the results of TARFVAE have been provided. There is no performance comparisons with other models. Intuitively, the experimental comparisons in CRPS should be the core experiments to demonstrate the effectiveness of your proposed generative model.\n\nQUESTIONS:\n- Please explain why you only compare with point forecasting models and mainly employ the point estimation metrics.\n- Please add necessary related studies in studying generative time-series forecasting models and comparing point and probabilistic forecasting paradigms.\n- Please add necessary experimental results to support your core claims.\n\nLIMITATIONS:\nSee the weakness part.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-05"}