5:15 PM - 7:15 PM
[SCG60-P11] Nonlinear friction model of rocks and crust based on rheology model by neural networks
Keywords:neural network, interpretability, friction model, rheology model
Numerous friction models have been considered in solid earth science to understand fault behavior. One classical model is the so-called spring-block model, which combines a spring and a block, and another model incorporates a dashpot to account for viscoelastic deformation (e.g., Yoshino, 1998). In this study, such models are referred to as rheological friction models. On the other hand, friction models using neural networks have also been devised in recent years, such as studies using physics-informed neural networks (PINNs) (e.g., Li et al., 2024).
Both seismic phenomena and neural networks are complex systems consisting of multiple elements, and it has been pointed out that they have in common the threshold nature of elements and the interaction between elements (Amari, 1987). Therefore, it is believed that by focusing on this commonality in friction models using neural networks, models that lead to a more essential understanding can be constructed. However, most recent studies using neural networks have focused on their use as a tool, focusing on the error back propagation method, which is a powerful parameter estimation method (e.g., Rouet-Leduc et al., 2017). Therefore, in this study, we investigated the mathematical equivalence between a rheological friction model, which has viscoelastic deformations in each element and elastic interactions between elements, and neural networks.
As a result, it became clear that the rheological friction model can be represented by a neural network with a recurrent structure of internal data. The model consists of a gate structure to determine block slippage and a reset structure to control the internal state. The model can describe spatio-temporal changes in the stress and strain state of a fault plane under a certain strain rate, or a spatio-temporal pattern of slip. If the model is regarded as a model that describes the relationship between strain rate and slip distribution, for example, it is possible to analyze the observed data as input and output. Furthermore, by preparing a data set consisting of strain rate and fault slip distribution, it is possible to consider an inverse analysis that can estimate the distribution of elastic modulus and surrounding viscosity of the fault surface from the error back propagation method.
In addition, the formulas in the neural network were organized by limit operations and polynomial approximations. As a result, it became clear that when the sampling interval was set close to zero, each weight of the neural network was determined by the coefficients for the highest order derivatives of stress and strain in the constitutive equation and the binomial coefficient. These coefficients represent the incremental relationship between stress and strain or strain rate. From this, the parameter fitting by error back propagation method is interpreted as finding this incremental relationship in continuous data from the data.
Since the neural network treated in this study consists of four elements: input, regression, interaction, and output, we can also consider a contrasting relationship with existing physical equations. For example, in Ruina's (1983) velocity-state-dependent friction constitutive law, the slip velocity corresponds to the input, the state variable to the regression, and the friction coefficient to the output. Also, in Terzaghi's (1924) equation describing one-dimensional consolidation phenomena, the pore water pressure at the next time is explained by regression and interaction.
Finally, this research model represents frictional behavior, a complex system with interactions, as a neural network, a complex system of neurons. This is a new example of a contrast between a physical system and a neural network, similar to Hopfield (1982), who pointed out the similarity between spin glass and neural networks. The contrast between the two is considered important in considering the interpretability of neural networks and the meaning of the error back propagation method.
Both seismic phenomena and neural networks are complex systems consisting of multiple elements, and it has been pointed out that they have in common the threshold nature of elements and the interaction between elements (Amari, 1987). Therefore, it is believed that by focusing on this commonality in friction models using neural networks, models that lead to a more essential understanding can be constructed. However, most recent studies using neural networks have focused on their use as a tool, focusing on the error back propagation method, which is a powerful parameter estimation method (e.g., Rouet-Leduc et al., 2017). Therefore, in this study, we investigated the mathematical equivalence between a rheological friction model, which has viscoelastic deformations in each element and elastic interactions between elements, and neural networks.
As a result, it became clear that the rheological friction model can be represented by a neural network with a recurrent structure of internal data. The model consists of a gate structure to determine block slippage and a reset structure to control the internal state. The model can describe spatio-temporal changes in the stress and strain state of a fault plane under a certain strain rate, or a spatio-temporal pattern of slip. If the model is regarded as a model that describes the relationship between strain rate and slip distribution, for example, it is possible to analyze the observed data as input and output. Furthermore, by preparing a data set consisting of strain rate and fault slip distribution, it is possible to consider an inverse analysis that can estimate the distribution of elastic modulus and surrounding viscosity of the fault surface from the error back propagation method.
In addition, the formulas in the neural network were organized by limit operations and polynomial approximations. As a result, it became clear that when the sampling interval was set close to zero, each weight of the neural network was determined by the coefficients for the highest order derivatives of stress and strain in the constitutive equation and the binomial coefficient. These coefficients represent the incremental relationship between stress and strain or strain rate. From this, the parameter fitting by error back propagation method is interpreted as finding this incremental relationship in continuous data from the data.
Since the neural network treated in this study consists of four elements: input, regression, interaction, and output, we can also consider a contrasting relationship with existing physical equations. For example, in Ruina's (1983) velocity-state-dependent friction constitutive law, the slip velocity corresponds to the input, the state variable to the regression, and the friction coefficient to the output. Also, in Terzaghi's (1924) equation describing one-dimensional consolidation phenomena, the pore water pressure at the next time is explained by regression and interaction.
Finally, this research model represents frictional behavior, a complex system with interactions, as a neural network, a complex system of neurons. This is a new example of a contrast between a physical system and a neural network, similar to Hopfield (1982), who pointed out the similarity between spin glass and neural networks. The contrast between the two is considered important in considering the interpretability of neural networks and the meaning of the error back propagation method.