TimeXer: Empowering Transformers for Time Series Forecasting with Exogenous Variables

Recent studies have demonstrated remarkable performance in time series forecasting. However, due to the partially-observed nature of real-world applications, solely focusing on the target of interest, so-called endogenous variables, is usually insufficient to guarantee accurate forecasting. Notably, a system is often recorded into multiple variables, where the exogenous series can provide valuable external information for endogenous variables. Thus, unlike prior well-established multivariate or univariate forecasting that either treats all the variables equally or overlooks exogenous information, this paper focuses on a practical setting, which is time series forecasting with exogenous variables. We propose a novel framework, TimeXer, to utilize external information to enhance the forecasting of endogenous variables. With a deftly designed embedding layer, TimeXer empowers the canonical Transformer architecture with the ability to reconcile endogenous and exogenous information, where patch-wise self-attention and variate-wise cross-attention are employed. Moreover, a global endogenous variate token is adopted to effectively bridge the exogenous series into endogenous temporal patches. Experimentally, TimeXer significantly improves time series forecasting with exogenous variables and achieves consistent state-of-the-art performance in twelve real-world forecasting benchmarks.

TimeSiam: A Pre-Training Framework for Siamese Time-Series Modeling

Time series pre-training has recently garnered wide attention for its potential to reduce labeling expenses and benefit various downstream tasks. Prior methods are mainly based on pre-training techniques well-acknowledged in vision or language, such as masked modeling and contrastive learning. However, randomly masking time series or calculating series-wise similarity will distort or neglect inherent temporal correlations crucial in time series data. To emphasize temporal correlation modeling, this paper proposes TimeSiam as a simple but effective self-supervised pre-training framework for Time series based on Siamese networks. Concretely, TimeSiam pre-trains Siamese encoders to capture intrinsic temporal correlations between randomly sampled past and current subseries. With a simple data augmentation method (e.g.~masking), TimeSiam can benefit from diverse augmented subseries and learn internal time-dependent representations through a past-to-current reconstruction. Moreover, learnable lineage embeddings are also introduced to distinguish temporal distance between sampled series and further foster the learning of diverse temporal correlations. TimeSiam consistently outperforms extensive advanced pre-training baselines, demonstrating superior forecasting and classification capabilities across 13 standard benchmarks in both intra- and cross-domain scenarios.

EuLagNet: Eulerian Fluid Prediction with Lagrangian Dynamics

Accurately predicting the future fluid is important to extensive areas, such as meteorology, oceanology and aerodynamics. However, since the fluid is usually observed from an Eulerian perspective, its active and intricate dynamics are seriously obscured and confounded in static grids, bringing horny challenges to the prediction. This paper introduces a new Lagrangian-guided paradigm to tackle the tanglesome fluid dynamics. Instead of solely predicting the future based on Eulerian observations, we propose the Eulerian-Lagrangian Dual Recurrent Network (EuLagNet), which captures multiscale fluid dynamics by tracking movements of adaptively sampled key particles on multiple scales and integrating dynamics information over time. Concretely, a EuLag Block is presented to communicate the learned Eulerian and Lagrangian features at each moment and scale, where the motion of tracked particles is inferred from Eulerian observations and their accumulated dynamics information is incorporated into Eulerian fields to guide future prediction. Tracking key particles not only provides a clear and interpretable clue for fluid dynamics but also makes our model free from modeling complex correlations among massive grids for better efficiency. Experimentally, EuLagNet excels in three challenging fluid prediction tasks, covering both 2D and 3D, simulated and real-world fluids.

Transolver: A Fast Transformer Solver for PDEs on General Geometries

Transformers have empowered many milestones across various fields and have recently been applied to solve partial differential equations (PDEs). However, since PDEs are typically discretized into large-scale meshes with complex geometries, it is challenging for Transformers to capture intricate physical correlations directly from massive individual points. Going beyond superficial and unwieldy meshes, we present Transolver based on a more foundational idea, which is learning intrinsic physical states hidden behind discretized geometries. Specifically, we propose a new Physics-Attention to adaptively split the discretized domain into a series of learnable slices of flexible shapes, where mesh points under similar physical states will be ascribed to the same slice. By calculating attention to physics-aware tokens encoded from slices, Transovler can effectively capture intricate physical correlations under complex geometrics, which also empowers the solver with endogenetic geometry-general modeling capacity and can be efficiently computed in linear complexity. Transolver achieves consistent state-of-the-art with 22\% relative gain across six standard benchmarks and also excels in large-scale industrial simulations, including car and airfoil designs.

HelmSim: Learning Helmholtz Dynamics for Interpretable Fluid Simulation

Fluid simulation is a long-standing challenge due to the intrinsic high-dimensional non-linear dynamics. Previous methods usually utilize the non-linear modeling capability of deep models to directly estimate velocity fields for future prediction. However, skipping over inherent physical properties but directly learning superficial velocity fields will overwhelm the model from generating precise or physics-reliable results. In this paper, we propose the HelmSim toward an accurate and interpretable simulator for fluid. Inspired by the Helmholtz theorem, we design a HelmDynamic block to learn the Helmholtz dynamics, which decomposes fluid dynamics into more solvable curl-free and divergence-free parts, physically corresponding to potential and stream functions of fluid. By embedding the HelmDynamic block into a Multiscale Integration Network, HelmSim can integrate learned Helmholtz dynamics along temporal dimension in multiple spatial scales to yield future fluid. Comparing with previous velocity estimating methods, HelmSim is faithfully derived from Helmholtz theorem and ravels out complex fluid dynamics with physically interpretable evidence. Experimentally, our proposed HelmSim achieves the consistent state-of-the-art in both numerical simulated and real-world observed benchmarks, even for scenarios with complex boundaries.

iTransformer: Inverted Transformers Are Effective for Time Series Forecasting

The recent boom of linear forecasting models questions the ongoing passion for architectural modifications of Transformer-based forecasters. These forecasters leverage Transformers to model the global dependencies over temporal tokens of time series, with each token formed by multiple variates of the same timestamp. However, Transformer is challenged in forecasting series with larger lookback windows due to performance degradation and computation explosion. Besides, the unified embedding for each temporal token fuses multiple variates with potentially unaligned timestamps and distinct physical measurements, which may fail in learning variate-centric representations and result in meaningless attention maps. In this work, we reflect on the competent duties of Transformer components and repurpose the Transformer architecture without any adaptation on the basic components. We propose iTransformer that simply inverts the duties of the attention mechanism and the feed-forward network. Specifically, the time points of individual series are embedded into variate tokens which are utilized by the attention mechanism to capture multivariate correlations; meanwhile, the feed-forward network is applied for each variate token to learn nonlinear representations. The iTransformer model achieves consistent state-of-the-art on several real-world datasets, which further empowers the Transformer family with promoted performance, generalization ability across different variates, and better utilization of arbitrary lookback windows, making it a nice alternative as the fundamental backbone of time series forecasting.

SimMTM: A Simple Pre-Training Framework for Masked Time-Series Modeling

Time series analysis is widely used in extensive areas. Recently, to reduce labeling expenses and benefit various tasks, self-supervised pre-training has attracted immense interest. One mainstream paradigm is masked modeling, which successfully pre-trains deep models by learning to reconstruct the masked content based on the unmasked part. However, since the semantic information of time series is mainly contained in temporal variations, the standard way of randomly masking a portion of time points will ruin vital temporal variations of time series seriously, making the reconstruction task too difficult to guide representation learning. We thus present SimMTM, a Simple pre-training framework for Masked Time-series Modeling. By relating masked modeling to manifold learning, SimMTM proposes to recover masked time points by the weighted aggregation of multiple neighbors outside the manifold, which eases the reconstruction task by assembling ruined but complementary temporal variations from multiple masked series. SimMTM further learns to uncover the local structure of the manifold helpful for masked modeling. Experimentally, SimMTM achieves state-of-the-art fine-tuning performance in two canonical time series analysis tasks: forecasting and classification, covering both in- and cross-domain settings.

Solving High-Dimensional PDEs with Latent Spectral Models

Deep models have achieved impressive progress in solving partial differential equations (PDEs). A burgeoning paradigm is learning neural operators to approximate the input-output mappings of PDEs. While previous deep models have explored the multiscale architectures and elaborative operator designs, they are limited to learning the operators as a whole in the coordinate space. In real physical science problems, PDEs are complex coupled equations with numerical solvers relying on discretization into high-dimensional coordinate space, which cannot be precisely approximated by a single operator nor efficiently learned for the curse of dimensionality. We present Latent Spectral Models (LSM) toward an efficient and precise solver for high-dimensional PDEs. Going beyond the coordinate space, LSM enables an attention-based hierarchical projection network to reduce the high-dimensional data into a compact latent space in linear time. Inspired by classical spectral methods in numerical analysis, we design a neural spectral block to solve PDEs in the latent space that well approximates complex input-output mappings via learning multiple basis operators, enjoying nice theoretical guarantees for convergence and approximation. Experimentally, LSM achieves consistent state-of-the-art and yields a relative gain of 11.5% averaged on seven benchmarks covering both solid and fluid physics.

TimesNet: Temporal 2D-Variation Modeling for General Time Series Analysis

Time series analysis is of immense importance in extensive applications, such as weather forecasting, anomaly detection, and action recognition. This paper focuses on temporal variation modeling, which is the common key problem of extensive analysis tasks. Previous methods attempt to accomplish this directly from the 1D time series, which is extremely challenging due to the intricate temporal patterns. Based on the observation of multi-periodicity in time series, we ravel out the complex temporal variations into the multiple intraperiod- and interperiod-variations. To tackle the limitations of 1D time series in representation capability, we extend the analysis of temporal variations into the 2D space by transforming the 1D time series into a set of 2D tensors based on multiple periods. This transformation can embed the intraperiod- and interperiod-variations into the columns and rows of the 2D tensors respectively, making the 2D-variations to be easily modeled by 2D kernels. Technically, we propose the TimesNet with TimesBlock as a task-general backbone for time series analysis. TimesBlock can discover the multi-periodicity adaptively and extract the complex temporal variations from transformed 2D tensors by a parameter-efficient inception block. Our proposed TimesNet achieves consistent state-of-the-art in five mainstream time series analysis tasks, including short- and long-term forecasting, imputation, classification, and anomaly detection.