LINOCS: Lookahead Inference of Networked Operators for Continuous Stability

TL;DR

LINOCS addresses the challenge of unstable long-horizon dynamics identification from noisy time-series by embedding lookahead constraints into operator inference. It unifies TI-LDS, SLDS, dLDS, and LTV into a single framework that optimizes adaptive weighted sums of multi-step predictions, e.g., minimize $ \sum_{k=0}^K w_k \| \tilde{x}_{t+1} - A_t \hat{x}_t^k \|_F^2 $. It demonstrates ground-truth operator recovery in synthetic data and robust long-term predictions in real neural recordings, outperforming 1-step baselines and DAD-type methods. The approach enhances interpretability and predictive stability for high-dimensional, non-stationary dynamical systems, with broad implications for neuroscience, ecology, and other complex domains.

Abstract

Identifying latent interactions within complex systems is key to unlocking deeper insights into their operational dynamics, including how their elements affect each other and contribute to the overall system behavior. For instance, in neuroscience, discovering neuron-to-neuron interactions is essential for understanding brain function; in ecology, recognizing the interactions among populations is key for understanding complex ecosystems. Such systems, often modeled as dynamical systems, typically exhibit noisy high-dimensional and non-stationary temporal behavior that renders their identification challenging. Existing dynamical system identification methods often yield operators that accurately capture short-term behavior but fail to predict long-term trends, suggesting an incomplete capture of the underlying process. Methods that consider extended forecasts (e.g., recurrent neural networks) lack explicit representations of element-wise interactions and require substantial training data, thereby failing to capture interpretable network operators. Here we introduce Lookahead-driven Inference of Networked Operators for Continuous Stability (LINOCS), a robust learning procedure for identifying hidden dynamical interactions in noisy time-series data. LINOCS integrates several multi-step predictions with adaptive weights during training to recover dynamical operators that can yield accurate long-term predictions. We demonstrate LINOCS' ability to recover the ground truth dynamical operators underlying synthetic time-series data for multiple dynamical systems models (including linear, piece-wise linear, time-changing linear systems' decomposition, and regularized linear time-varying systems) as well as its capability to produce meaningful operators with robust reconstructions through various real-world examples.

Figures (22)

• Figure 1: Illustration of the problem and approach.A: Models that perform well on $1$-step prediction (i.e., prediction order $0$, in red) , often fail in higher orders reconstruction (e.g., order-$T$, in green). B: LINOCS (e.g., for training order $K=3$) integrates weighted multi-step reconstructions for all $k = 0 \dots K$ orders. It adapts the weights of these reconstruction orders to prioritize minimizing large errors at lower orders before addressing higher orders. The system gradually increases the weight of the most effective lookahead reconstruction until convergence conditions are met. C: LINOCS improves long-term reconstruction during training iterations. D: Weights of different lookahead training orders ($k$). $w_k: \mathbb{R}^2 \rightarrow \mathbb{R}$ is a function of the order $k$ and the $k$-th order reconstruction error $e$. Top: Illustration of an exemplary effect of $e$ on $w$ when fixing $k$. Bottom: Illustration of an exemplary effect of $k$ on $w$ when fixing $e$.
• Figure 2: Example time-varying LDS systemsA: The baseline linear time-invariant dynamical system will present constant dynamical operator constant over time. B: Switching linear dynamical systems (SLDS) jump between different linear operators that are time-invariant between jumps. C: "Pseudo-Switching" dynamics is similar to SLDS with the inclusion of smoother transitions between periods of constant linear dynamics. D: The decomposed linear dynamical systems (dLDS) model is a generalization of SLDS to sparse time-changing linear combinations of linear operators. dLDS can also model negative coefficients.
• Figure 3: Linear System Experiment.A: Real vs. identified operators and offsets. B: Quiver plots of real and identified operators. C: Highlighted differences in effects between real operators and inferred operators showing how small differences in dynamic operators gain prominence during lookahead reconstruction (calculation details in Section \ref{['sec:cal_details_linear_diff']}). D: Full lookahead reconstructions (ground truth vs. baselines) show swift convergence to the origin for the 1-step optimization (yellow) and divergence for DAD-based results (three most right subplots). E: Frobenius norm of the differences between the ground truth operators ($\bm{A}$) and the identified operators ($\widehat{\bm{A}}$). F: MSE under increasing prediction orders. For all orders, LINOCS achieves better (lower) MSE compared to 1-step optimization. G: Full lookahead post-training predictions using operators identified by 1-step optimization (yellow) vs. the predictions using those identified by LINOCS (blue) under various training orders (rows) and prediction orders (columns). H, I: LINOCS reconstruction compared to 1-step optimization under increasing noise levels demonstrates its robustness. J: Propagating the identified operators until reaching a relative reconstruction error of $\sim 1$. LINOCS identifies operators that can accurately predict $\sim$ 35,000 time points, much higher than 1-step training that decay immediately.
• Figure 4: Results on switching systems.A: Active discrete states for LINOCS (blue) compared to baselines, including SLDS and rSLDS with 10 or 100 training epochs. B: Correlation between the ground truth dynamics ($\bm{x}$) and the full-lookahead reconstructed dynamics ($\widehat{\bm{x}}$). C: Correlation between the ground truth operators ($\bm{f}$) and the identified operators ($\widehat{\bm{f}}$). D: Correlation between the ground truth coefficients ($\bm{c}$) and the identified coefficients ($\widehat{\bm{c}}$). E: Difference between the ground-truth sub-dynamics ($\widehat{\bm{F}}$) and reconstructed basis dynamics by different models. LINOCS was able to achieve sub-dynamics that are much closer to the ground truth than the other baselines.
• Figure 5: Decomposed linear dynamical systems results.A: Ground truth dynamics compared to 1-step (top) and full lookahead (bottom) reconstructions for non-LINOCS dLDS ($K_{train} = 1$) and LINOCS-dLDS with different training orders. B: MSE (pink) and correlation (green) between ground truth operators and the operators identified by LINOCS under different orders. C: MSE (pink) and correlation (green) between ground truth dynamics and full lookahead reconstructions using the different LINOCS training orders. D: Local MSE for 1-step (left) and for full lookahead reconstruction (right) over the time points of the dynamics.