A new approach to data-driven modeling of complex spatiotemporal dynamical systems

Recently, Liu Yang, a long-term associate professor at the School of Engineering Science of the University of Chinese Academy of Sciences, published research results entitled Encoding physics to learn reaction-diffusion process in Nature Machine Intelligence. This study proposes a new deep learning architecture for embedding physical knowledge, namely PeRCNN, for partial differential equation (PDE) forward and negative problem solving, nonlinear spatiotemporal dynamical system modeling and control equation discovery, aiming to improve the accuracy and interpretability of modeling of complex spatiotemporal dynamical systems based on sparse and noisy data. This work establishes a cyclic convolutional neural network framework for forced coding of physical structures, which effectively solves the dependence of the network on training data, breaks through the unexplainable bottleneck of the neural network model, improves the extrapolation and generalization of the model, and realizes verification in various reaction diffusion system (RD) problem scenarios.

In general, modeling and simulating complex space-time dynamical systems is a challenging task due to limited prior knowledge and the difficulty of describing nonlinear processes of system variables with clear partial differential equation (PDE) formulas. Common machine learning methods rely on a large amount of training data, and there are basic scientific problems such as poor interpretability, weak generalization, and uncontrollable errors. Prior physics knowledge (such as control equations) is added to deep learning as constraints to enhance the interpretability of the model and alleviate data dependence to a certain extent. However, existing deep learning methods based on physical information often require soft constraints to impose physical laws, and model performance depends largely on the appropriate setting of hyperparameters. Therefore, there is an urgent need to develop new knowledge-embedded learning models to capture the underlying spatiotemporal dynamic evolution mechanism in the data.

The researchers proposed a new learning model, the physically encoded recurrent convolutional neural network (PeRCNN), as shown in Figure 1. The main advantage of PeRCNN is that it can encode the structure of prior physical knowledge into the network, using the spatiotemporal learning paradigm, aiming to establish a general and robust learning model, which can ensure that the obtained network strictly obeys the given prior physical knowledge (such as PDE structure, initial and boundary conditions), making the network interpretable. The model encodes a given physical structure through a recurrent convolutional neural network, improving the ability to learn complex spatiotemporal dynamics under sparse and noisy data.

Figure 1. PeRCNN model diagram

Through a large number of numerical experiments, this work demonstrates the effectiveness of PeRCNN’s analysis of the forward (Figure 2) and reverse (Figure 3) problem of reaction-diffusion partial differential equations. Comparison with several baseline models shows that the physical coding learning paradigm has unique extrapolation ability, generalization ability and robustness to data noise or sparse data. In data-driven simulation experiments, PeRCNN achieved SOTA results; PeRCNN’s flat error propagation curve (Figure 4) proves PeRCNN’s remarkable extrapolation and generalization capabilities, which means that the method can not only accurately predict the evolution of complex spatiotemporal dynamical systems, but also capture the physical mechanism behind the model, and show a certain versatility when dealing with new working conditions.

Figure 2. Experimental results (PDE positive questions)

Figure 3. Experimental results (PDE inverse problem)

Figure 4. Experimental results (data-driven modeling, error propagation and extrapolation diagrams)

In addition, this work combines PeRCNN with sparse regression algorithms to solve potential PDE discovery problems (Figure 5), which can further extract analytic expressions that control the underlying physical mechanisms in the learning model. The coupling scheme enables the model to iteratively optimize the network parameters, fine-tune the discovered PDE structure and coefficients, obtain the simple expression of the final PDE, and accurately and reliably discover the underlying physical laws in the sparse and noisy measurement data.

Figure 5. Control partial differential equation discovery flowchart

This study demonstrates the effectiveness of PeRCNN on various reaction-diffusion systems, but the model is theoretically applicable to other types of space-time dynamical systems, such as the two-dimensional Burgers equation with convective term and the Kolmogorov turbulence (NS equation) with a Reynolds number of 1000.

These results bring new advances in the field of data-driven modeling of complex spatiotemporal dynamical systems, providing scientists and engineers with more powerful tools to explore and predict natural and engineering phenomena. This approach, which combines deep learning and prior physics, is expected to play an important role in applications in disciplines such as fluid mechanics, biochemistry, environmental science, engineering, and materials science. (Source: University of Chinese Academy of Sciences)

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