Long-term time-series forecasting (LTSF) models are often presented as general-purpose solutions that can be applied across domains, implicitly assuming that all data is pointwise predictable. Using chaotic systems such as Lorenz-63 as a case study, we argue that geometric structure - not pointwise prediction - is the right abstraction for a dynamic-agnostic foundational model. Minimizing the Wasserstein-2 distance (W2), which captures geometric changes, and providing a spectral view of dynamics are essential for long-horizon forecasting. Our model, FRIREN (Flow-inspired Representations via Interpretable Eigen-networks), implements an augmented normalizing-flow block that embeds data into a normally distributed latent representation. It then generates a W2-efficient optimal path that can be decomposed into rotation, scaling, inverse rotation, and translation. This architecture yields locally generated, geometry-preserving predictions that are independent of the underlying dynamics, and a global spectral representation that functions as a finite Koopman operator with a small modification. This enables practitioners to identify which modes grow, decay, or oscillate, both locally and system-wide. FRIREN achieves an MSE of 11.4, MAE of 1.6, and SWD of 0.96 on Lorenz-63 in a 336-in, 336-out, dt=0.01 setting, surpassing TimeMixer (MSE 27.3, MAE 2.8, SWD 2.1). The model maintains effective prediction for 274 out of 336 steps, approximately 2.5 Lyapunov times. On Rossler (96-in, 336-out), FRIREN achieves an MSE of 0.0349, MAE of 0.0953, and SWD of 0.0170, outperforming TimeMixer's MSE of 4.3988, MAE of 0.886, and SWD of 3.2065. FRIREN is also competitive on standard LTSF datasets such as ETT and Weather. By connecting modern generative flows with classical spectral analysis, FRIREN makes long-term forecasting both accurate and interpretable, setting a new benchmark for LTSF model design.

翻译：长期时间序列预测（LTSF）模型常被呈现为跨领域通用的解决方案，其隐含假设所有数据均可逐点预测。本文以Lorenz-63等混沌系统为例，论证几何结构（而非逐点预测）才是构建动态无关基础模型的正确抽象。最小化捕捉几何变化的Wasserstein-2距离（W2），并提供动态系统的光谱视角，对于长期预测至关重要。我们提出的FRIREN模型（通过可解释特征网络实现的流启发表征）采用增强型归一化流模块，将数据嵌入正态分布的潜在表征，继而生成可通过旋转、缩放、逆旋转和平移分解的W2最优路径。该架构既能生成与底层动态无关的局部几何保持预测，又能通过小幅修正构建起类似有限Koopman算子的全局光谱表征，使研究者能够识别局部和全局范围内哪些模态在增长、衰减或振荡。在Lorenz-63系统（输入336步，输出336步，dt=0.01）中，FRIREN取得MSE 11.4、MAE 1.6、SWD 0.96，显著超越TimeMixer（MSE 27.3、MAE 2.8、SWD 2.1），在336步预测中有效保持274步（约2.5个Lyapunov时间）。在Rossler系统（输入96步，输出336步）中，FRIREN以MSE 0.0349、MAE 0.0953、SWD 0.0170大幅优于TimeMixer的MSE 4.3988、MAE 0.886、SWD 3.2065。该模型在ETT、Weather等标准LTSF数据集上也展现出竞争力。通过将现代生成流与经典光谱分析相结合，FRIREN实现了兼具准确性与可解释性的长期预测，为LTSF模型设计树立了新基准。