Jiecheng Lu

Jiecheng Lu, Xu Han, Shihao Yang

Long-term time series forecasting (LTSF) is important for various domains but is confronted by challenges in handling the complex temporal-contextual relationships. As multivariate input models underperforming some recent univariate counterparts, we posit that the issue lies in the inefficiency of existing multivariate LTSF Transformers to model series-wise relationships: the characteristic differences between series are often captured incorrectly. To address this, we introduce ARM: a multivariate temporal-contextual adaptive learning method, which is an enhanced architecture specifically designed for multivariate LTSF modelling. ARM employs Adaptive Univariate Effect Learning (AUEL), Random Dropping (RD) training strategy, and Multi-kernel Local Smoothing (MKLS), to better handle individual series temporal patterns and correctly learn inter-series dependencies. ARM demonstrates superior performance on multiple benchmarks without significantly increasing computational costs compared to vanilla Transformer, thereby advancing the state-of-the-art in LTSF. ARM is also generally applicable to other LTSF architecture beyond vanilla Transformer.

Jiecheng Lu, Jieqi Di, Runhua Wu et al.

Many decision-facing stochastic systems are observed through aggregate distributions rather than scalar trajectories: queue occupancies, mobility shares, public-health mixtures, generation-source shares, ecological compositions, and air-quality severity profiles all live on the probability simplex and evolve over time. We study causal (online) forecasting for these distribution-valued time series and argue that the transition operator itself should be structured around the simplex. We introduce CAST (Causal Anchored Simplex Transport), a successor-local operator that (i) retrieves empirical successors from causal context, (ii) stabilizes them with a persistence anchor, and (iii) applies a bounded local stochastic transport on ordered supports; every stage preserves the simplex by construction. We identify a structural failure mode, latent transition-kernel aliasing, where similar observed distributions evolve differently under different contextual regimes, and prove that any forecaster depending only on an aliased summary incurs an irreducible weighted Jensen-Shannon excess-risk lower bound, while the CAST hypothesis class contains the regime-aware Bayes successor; for ordered supports an additional Pinsker separation holds whenever the transported successor lies outside the no-transport anchor hull. On eleven public and simulated benchmarks spanning ecology, energy, diet, mortality, employment, air quality, severe weather, mobility, and G/G/1, G_t/G/1 queue occupancy, CAST attains the best average rank on both one-step KL (1.27) and autoregressive rollout JSD (1.91), winning 8/11 sections on each metric against a broad statistical, compositional, recurrent, convolutional, and Transformer baseline set, and top-2 on all 11 sections for offline KL. Component ablations and a controlled synthetic aliasing experiment corroborate the theory.

Viresh Pati, Yubin Kim, Vinh Pham et al.

This paper presents $\textbf{CAPS}$ (Clock-weighted Aggregation with Prefix-products and Softmax), a structured attention mechanism for time series forecasting that decouples three distinct temporal structures: global trends, local shocks, and seasonal patterns. Standard softmax attention entangles these through global normalization, while recent recurrent models sacrifice long-term, order-independent selection for order-dependent causal structure. CAPS combines SO(2) rotations for phase alignment with three additive gating paths -- Riemann softmax, prefix-product gates, and a Clock baseline -- within a single attention layer. We introduce the Clock mechanism, a learned temporal weighting that modulates these paths through a shared notion of temporal importance. Experiments on long- and short-term forecasting benchmarks surpass vanilla softmax and linear attention mechanisms and demonstrate competitive performance against seven strong baselines with linear complexity. Our code implementation is available at https://github.com/vireshpati/CAPS-Attention.

Jevon Twitty, Vinh Pham, Nitiwith Rotchanarak et al.

A persistent paradox in time-series forecasting is that structurally simple MLP and linear models often outperform high-capacity Transformers. We argue that this gap arises from a mismatch in the sequence-modeling primitive: while many time-series dynamics are governed by global temporal operators (e.g., filtering and harmonic structure), standard attention forms each output as a convex combination of inputs. This restricts its ability to represent signed and oscillatory transformations that are fundamental to temporal signal processing. We formalize this limitation as a simplex-constrained mixing bottleneck in softmax attention, which becomes especially restrictive for operator-driven time-series tasks. To address this, we propose $\textbf{Temporal Operator Attention (TOA)}$, a framework that augments attention with explicit, learnable sequence-space operators, enabling direct signed mixing across time while preserving input-dependent adaptivity. To make dense $N \times N$ operators practical, we introduce Stochastic Operator Regularization, a high-variance dropout mechanism that stabilizes training and prevents trivial memorization. Across forecasting, anomaly detection, and classification benchmarks, TOA consistently improves performance when integrated into standard backbones such as PatchTST and iTransformer, with particularly strong gains in reconstruction-heavy tasks. These results suggest that explicit operator learning is a key ingredient for effective time-series modeling.

Jiecheng Lu, Shihao Yang

Self-attention is often viewed as probabilistic query-key lookup, motivating designs that preserve normalized attention scores and fixed positional semantics. We advocate a simpler and more unified perspective: an autoregressive attention head can be viewed as a dynamic two-layer MLP whose weights are instantiated from the context history. From this view, attention scores form an ever-growing hidden representation, and standard MLP activations such as ReLU or GLU naturally implement input-conditioned selection over a context-dependent memory pool rather than a probability distribution. Based on this formulation, we introduce HyperMLP and HyperGLU, which learn dynamic mixing in both feature space and sequence space, using a reverse-offset (lag) layout to align temporal mixing with autoregressive semantics. We provide theoretical characterizations of the expressivity and implications of this structure, and empirically show that HyperMLP/HyperGLU consistently outperform strong softmax-attention baselines under matched parameter budgets.

Jiecheng Lu, Xu Han, Yan Sun et al.

Linear attention methods offer Transformers $O(N)$ complexity but typically underperform standard softmax attention. We identify two fundamental limitations affecting these approaches: the restriction to convex combinations that only permits additive information blending, and uniform accumulated weight bias that dilutes attention in long contexts. We propose Zero-Sum Linear Attention (ZeroS), which addresses these limitations by removing the constant zero-order term $1/t$ and reweighting the remaining zero-sum softmax residuals. This modification creates mathematically stable weights, enabling both positive and negative values and allowing a single attention layer to perform contrastive operations. While maintaining $O(N)$ complexity, ZeroS theoretically expands the set of representable functions compared to convex combinations. Empirically, it matches or exceeds standard softmax attention across various sequence modeling benchmarks.

Jiecheng Lu, Shihao Yang

Standard attention stores keys/values losslessly but reads them via a per-head convex average, blocking channel-wise selection. We propose the Free Energy Mixer (FEM): a free-energy (log-sum-exp) read that applies a value-driven, per-channel log-linear tilt to a fast prior (e.g., from queries/keys in standard attention) over indices. Unlike methods that attempt to improve and enrich the $(q,k)$ scoring distribution, FEM treats it as a prior and yields a value-aware posterior read at unchanged complexity, smoothly moving from averaging to per-channel selection as the learnable inverse temperature increases, while still preserving parallelism and the original asymptotic complexity ($O(T^2)$ for softmax; $O(T)$ for linearizable variants). We instantiate a two-level gated FEM that is plug-and-play with standard and linear attention, linear RNNs and SSMs. It consistently outperforms strong baselines on NLP, vision, and time-series at matched parameter budgets.

Yubin Kim, Viresh Pati, Jevon Twitty et al.

Transformer architectures have established strong baselines in time series forecasting, yet they typically rely on positional encodings that assume uniform, index-based temporal progression. However, real-world systems, from shifting financial cycles to elastic biological rhythms, frequently exhibit "time-warped" dynamics where the effective flow of time decouples from the sampling index. In this work, we first formalize this misalignment and prove that rotary position embedding (RoPE) is mathematically incapable of representing non-affine temporal warping. To address this, we propose Symplectic Positional Embeddings (SyPE), a learnable encoding framework derived from Hamiltonian mechanics. SyPE strictly generalizes RoPE by extending the rotation group $\mathrm{SO}(2)$ to the symplectic group $\mathrm{Sp}(2,\mathbb{R})$, modulated by a novel input-dependent adaptive warp module. By allowing the attention mechanism to adaptively dilate or contract temporal coordinates end-to-end, our approach captures locally varying periodicities without requiring pre-defined warping functions. We implement this mechanism in StretchTime, a multivariate forecasting architecture that achieves state-of-the-art performance on standard benchmarks, demonstrating superior robustness on datasets exhibiting non-stationary temporal dynamics.

Jiecheng Lu, Xu Han, Yan Sun et al.

For Multivariate Time Series Forecasting (MTSF), recent deep learning applications show that univariate models frequently outperform multivariate ones. To address the difficiency in multivariate models, we introduce a method to Construct Auxiliary Time Series (CATS) that functions like a 2D temporal-contextual attention mechanism, which generates Auxiliary Time Series (ATS) from Original Time Series (OTS) to effectively represent and incorporate inter-series relationships for forecasting. Key principles of ATS - continuity, sparsity, and variability - are identified and implemented through different modules. Even with a basic 2-layer MLP as core predictor, CATS achieves state-of-the-art, significantly reducing complexity and parameters compared to previous multivariate models, marking it an efficient and transferable MTSF solution.

Jiecheng Lu, Yan Sun, Shihao Yang

Recent Transformer-based large language models (LLMs) demonstrate in-context learning ability to perform various functions based solely on the provided context, without updating model parameters. To fully utilize the in-context capabilities in time series forecasting (TSF) problems, unlike previous Transformer-based or LLM-based time series forecasting methods, we reformulate "time series forecasting tasks" as input tokens by constructing a series of (lookback, future) pairs within the tokens. This method aligns more closely with the inherent in-context mechanisms, and is more parameter-efficient without the need of using pre-trained LLM parameters. Furthermore, it addresses issues such as overfitting in existing Transformer-based TSF models, consistently achieving better performance across full-data, few-shot, and zero-shot settings compared to previous architectures.

Jiecheng Lu, Shihao Yang

Autoregressive attention-based time series forecasting (TSF) has drawn increasing interest, with mechanisms like linear attention sometimes outperforming vanilla attention. However, deeper Transformer architectures frequently misalign with autoregressive objectives, obscuring the underlying VAR structure embedded within linear attention and hindering their ability to capture the data generative processes in TSF. In this work, we first show that a single linear attention layer can be interpreted as a dynamic vector autoregressive (VAR) structure. We then explain that existing multi-layer Transformers have structural mismatches with the autoregressive forecasting objective, which impair interpretability and generalization ability. To address this, we show that by rearranging the MLP, attention, and input-output flow, multi-layer linear attention can also be aligned as a VAR model. Then, we propose Structural Aligned Mixture of VAR (SAMoVAR), a linear Transformer variant that integrates interpretable dynamic VAR weights for multivariate TSF. By aligning the Transformer architecture with autoregressive objectives, SAMoVAR delivers improved performance, interpretability, and computational efficiency, comparing to SOTA TSF models.