Thank you for your response and support! We would be delighted to provide you with further clarification on the theory and approach choices.
* **Motivation and Significance**: Machine learning has historically been bifurcated into physics-driven approaches (such as PDE numerical solvers, PINN, etc.) and data-driven methods (like large models, various neural networks, neural operators, etc.). While data-driven techniques have excelled in time series modeling (as well as in computer vision, natural language processing, etc.), their black-box nature poses challenges for sustained advancement within the machine learning community. Recent publications in esteemed journals like Nature, Science, PNAS, etc., have showcased a fusion of physics-driven and data-driven methodologies, hinting at the emergence of a prominent research trend involving the integration of physics priors into deep learning models. **Our research marks the inaugural incorporation of PDE dynamical system knowledge into the realm of time series prediction, promising to facilitate interdisciplinary amalgamation and the creation of interpretable deep time series models.**
* **Dynamic System**: also known as PDE dynamical systems. Let the domain $S$ be an open subset of $\mathbb{R}^d$ and set an integer $k \geq 1$. Define the system state as $\boldsymbol{x}: S \mapsto \mathbb{R}^m$ where $\boldsymbol{x}=\left(x^1, \ldots, x^m\right)$. Then, an expression of the form:$\mathcal{F}\left(D^k \boldsymbol{x}(s), D^{k-1} \boldsymbol{x}(s), \ldots, D \boldsymbol{x}(s), \boldsymbol{x}(s), s\right)=0$ is called a $k^{\text {th }}$-order system of partial differential equation, where $\mathcal{F}: \mathbb{R}^{m d^k} \times \mathbb{R}^{m d^{k-1}} \times \ldots \times \mathbb{R}^{m d} \times \mathbb{R}^m \times S \mapsto \mathbb{R}^m$ and $s \in S$.
real-time series are observed values obtained from the system through a function h. By starting from dynamical systems instead of the original value, it becomes possible to have a better understanding of the fundamental temporal behavior. Moreover, this approach offers the advantages of interpretability and visualization.
* **Chaos Theory**: The study of chaos can be considered a branch of dynamical systems, focusing on the fact that both linear and nonlinear dynamical systems tend to exhibit certain fixed shapes in their trajectories, known as attractors. What may appear as irregular behavior in the time domain often reveals stable structures in the dynamical trajectories. The concept of attractors can readily be correlated with current deep-learning pattern recognition technologies.
* **Why SSM?**: The reason for choosing SSM to encode dynamical structures is that polynomials are widely recognized as universal approximators in dynamical systems research. The mathematical interpretation of SSM conveniently aligns with polynomial projection, making SSM well-suited for encoding dynamical structures. And we have introduced a novel SSM evolution matrix {-1, -1, -1, ...} to describe finite window approximations.
* **Why SSM + Multiple Orthogonal Subspaces**: Building upon SSM, we drew inspiration from chaos-related research (**Theorem 3 Attractor evolution error, line 163**) and proposed an enhanced version of SSM that utilizes distinct orthogonal subspaces to store various attractor structures. This innovation was validated through experimental results.
* **Why Frequency Evolution**: In the field of neuroscience research (where EEG and ECG are recognized as chaos datasets), it has been established that attractors are amplified in the frequency domain. Therefore, this paper adopts a frequency-domain evolution strategy. Besides, by discarding high-frequency components, noise effects can be mitigated, reducing computational complexity. Additionally, we explored the effects of Hopfield evolution and KNN clustering evolution, and through experimentation, **we found that these two methods were not as effective as frequency-domain evolution. We provided explanations for these findings in line 257**.
* **Follow-up Work**: In our recent research, we have discovered that **incorporating dynamical system priors leads to performance enhancements across all four types of time-series tasks (prediction, classification, interpolation, anomaly detection) and all four neural network architectures (convolutional, attention-based, linear, SSM). These improvements include a tenfold reduction in parameter count, as well as more stable gradients.** This work is set to be released soon—stay tuned for more updates.