Hybrid Forecasting of Chaotic Processes: Using Machine Learning in Conjunction with a Knowledge-Based Model (1803.04779v1)

Abstract: A model-based approach to forecasting chaotic dynamical systems utilizes knowledge of the physical processes governing the dynamics to build an approximate mathematical model of the system. In contrast, machine learning techniques have demonstrated promising results for forecasting chaotic systems purely from past time series measurements of system state variables (training data), without prior knowledge of the system dynamics. The motivation for this paper is the potential of machine learning for filling in the gaps in our underlying mechanistic knowledge that cause widely-used knowledge-based models to be inaccurate. Thus we here propose a general method that leverages the advantages of these two approaches by combining a knowledge-based model and a machine learning technique to build a hybrid forecasting scheme. Potential applications for such an approach are numerous (e.g., improving weather forecasting). We demonstrate and test the utility of this approach using a particular illustrative version of a machine learning known as reservoir computing, and we apply the resulting hybrid forecaster to a low-dimensional chaotic system, as well as to a high-dimensional spatiotemporal chaotic system. These tests yield extremely promising results in that our hybrid technique is able to accurately predict for a much longer period of time than either its machine-learning component or its model-based component alone.

• The paper introduces a hybrid forecasting model that fuses reservoir computing with physics-based models to extend prediction horizons in chaotic systems.
• It demonstrates that integrating data-driven and knowledge-based approaches enhances forecast accuracy and maintains robustness even when system dynamics are imperfectly modeled.
• Empirical tests on the Lorenz system and Kuramoto-Sivashinsky equations reveal prolonged valid predictions at reduced computational costs.

Overview of Hybrid Forecasting of Chaotic Processes: Using Machine Learning in Conjunction with a Knowledge-Based Model

This paper presents a method for combining machine learning techniques, specifically reservoir computing, with traditional knowledge-based models to forecast chaotic dynamical systems. The approach aims to leverage the complementary strengths of data-driven and model-based prediction methods to overcome the individual shortcomings associated with each. The authors demonstrate the efficacy of their hybrid approach through rigorous empirical testing on both low-dimensional and high-dimensional chaotic systems, including the Lorenz system and the Kuramoto-Sivashinsky equations.

The Hybrid Forecasting Model

The central innovation in this work is the development of a hybrid forecasting model that integrates reservoir computing—a machine learning technique known for its ability to model complex dynamics—into the framework of classical knowledge-based modeling. The knowledge-based model utilizes predefined physical laws and principles, while the reservoir computing component operates on historical observational data without inherently understanding the system's mechanics.

This combination allows the hybrid model to compensate for the inaccuracies commonly present in knowledge-based models due to incomplete understanding or erroneous assumptions about the underlying dynamics. Similarly, by incorporating mechanistic insight into the data-driven reservoir approach, the hybrid model can mitigate the need for extensive, computationally expensive datasets typically required for effective training in machine learning models.

Experimental Evaluation and Results

The paper includes extensive testing of the proposed hybrid approach using the Lorenz system and the Kuramoto-Sivashinsky equations. The Lorenz system serves as a benchmark for low-dimensional chaos, whereas the Kuramoto-Sivashinsky equations present a more challenging, high-dimensional spatiotemporal chaotic system.

Key findings include:

• Improvement in Forecasting Accuracy: The hybrid model demonstrates significantly improved prediction durations when compared to standalone knowledge-based or reservoir-only models. For example, the hybrid system can predict the Lorenz system's evolution accurately over more extended periods than either component acting alone.
• Robustness to Model Imperfections: The hybrid approach provides favorable outcomes even when knowledge-based models are deliberately degraded by introducing systematic errors, simulating practical scenarios where mechanistic models are imperfect.
• Computational Efficiency: The hybrid model displays potential to yield good prediction performance at reduced computational costs compared to using a much larger reservoir-only model. For instance, achieving similar valid prediction times with a smaller reservoir when used in conjunction with an imperfect knowledge-based model.

Practical and Theoretical Implications

From a practical perspective, this research holds promise for applications in areas such as weather forecasting, where chaotic dynamical systems are prevalent, and the prediction challenge is acute. The demonstrated reduction in computational demand could translate into cost savings and more accessible high-performance forecasting tools.

Theoretically, this work adds to the understanding of how machine learning models can be augmented by mechanistic insights to capture the intricate behaviors of chaotic systems more accurately. It advocates for an interdisciplinary approach, blending data-driven and model-based strategies, to address complex scientific problems.

Future Directions

The authors speculate on several potential future developments arising from their research. Expanding the hybrid approach to other types of machine learning models could reveal additional benefits or optimize existing methodologies. Moreover, exploring different implementations of reservoir computing, such as physical reservoir setups using optical or electronic circuits, may provide further gains in speed and efficiency.

In conclusion, this paper presents a compelling case for hybrid models in forecasting chaotic processes, thanks to the synergetic use of machine learning and knowledge-based strategies. The hybrid model not only improves the fidelity of predictions but also enhances the applicability and sustainability of forecasting solutions in various scientific and technological fields.