Sparse identification of nonlinear dynamics with library optimization mechanism: Recursive long-term prediction perspective

TL;DR

This work extends SINDy by introducing SINDy-LOM, which treats the library design as an optimization problem over parametrized basis functions. The method employs a two-layer architecture: an inner sparse regression that identifies a parsimonious state representation and an outer optimization that tunes the library parameters to maximize recursive long-term prediction accuracy. By incorporating RLT-focused criteria, SINDy-LOM yields more reliable multi-step forecasts and reduces user burden through automatic library design, demonstrated on a single-link robot and a diesel engine airpath system. The results show that the learned models are both interpretable (closed-form) and capable of accurate long-horizon predictions, highlighting the approach’s practical value for control-oriented data-driven modeling.

Abstract

The sparse identification of nonlinear dynamics (SINDy) approach can discover the governing equations of dynamical systems based on measurement data, where the dynamical model is identified as the sparse linear combination of the given basis functions. A major challenge in SINDy is the design of a library, which is a set of candidate basis functions, as the appropriate library is not trivial for many dynamical systems. To overcome this difficulty, this study proposes SINDy with library optimization mechanism (SINDy-LOM), which is a combination of the sparse regression technique and the novel learning strategy of the library. In the proposed approach, the basis functions are parametrized. The SINDy-LOM approach involves a two-layer optimization architecture: the inner-layer, in which the data-driven model is extracted as the sparse linear combination of the candidate basis functions, and the outer-layer, in which the basis functions are optimized from the viewpoint of the recursive long-term (RLT) prediction accuracy; thus, the library design is reformulated as the optimization of the parametrized basis functions. The dynamical model obtained by SINDy-LOM has good interpretability and usability, as this approach yields a parsimonious closed-form model. The library optimization mechanism significantly reduces user burden. The RLT perspective improves the reliability of the resulting model compared with the traditional SINDy approach that can only ensure the one-step-ahead prediction accuracy. The effectiveness of the proposed approach is verified through numerical experiments.

Figures (10)

• Figure 1: Schematic of the SINDy-LOM approach.
• Figure 2: Visualization of the resulting coefficient matrix $\varXi^{\ast} = \xi_{1}^{\ast}\xi_{2}^{\ast}$ obtained by Strategies #1--#3 (Example #1).
• Figure 3: RLT prediction results and errors for Example #1: (a) results for the SR data, (b) enlarged view of (a) in the time range $7400$--$7600$ steps, and (c) results for the oLL data. The black solid line represents the true data, whereas the blue dotted, green dash-dotted, and magenta broken lines represent the RLT predictions obtained from the models identified by Strategies #1--#3, respectively. The model derived from Strategy #3 achieved the highest prediction accuracy.
• Figure 4: RLT prediction results and errors obtained from the models identified by the proposed approach using noisy data (Example #1): (a) an enlarged view of the SR trajectory prediction over the time interval $7400$--$7600$ steps, and (b) the prediction for the oLL trajectory. The models identified from noisy training data demonstrated high RLT prediction accuracy.
• Figure 5: System to be modeled in Example #2: diesel engine airpath system.

Theorems & Definitions (7)

• Remark 1
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• Remark 5
• Remark 6
• Remark 7