特異値分解によるモード抽出を利用した関数同定

抄録

Many studies have been conducted to clarify unknown input/output systems based on measured data obtained via experiments and observations. Specifically, neural networks, which have been significantly developed in recent years, can be used for equation discovery. However, the identified networks are extremely large and difficult to understand. Although many studies have been conducted on mathematical expressions that are easy to understand, there is a paucity of studies on methods for determining a simple equation by eliminating unnecessary terms.

 

In this study, a novel method is proposed for identifying unknown equations by using mode extraction via singular value decomposition. This method is based on the modal iterative error correction method, which is effective for solving inverse problems with strong discontinuities. Additionally, the proposed method includes a process for removing null modes to obtain a simple equation without unnecessary terms.

 

The key findings of this study are as follows:

(1) We consider equation discovery as an inverse problem of unknowns such as coefficients and exponents. Thus, a novel method is proposed wherein the “process of improving the precision of an equation” and the “process of deleting unnecessary terms” are alternately iterated based on singular value decomposition results of Jacobian matrices of unknown parameters. This method enables the identification of functions consisting of only essential terms and ensures the reproducibility of unknown input/output systems.

 

(2) The applicability of the proposed equation discovery method is confirmed via a sample problem on cantilever deflection. Thus, an equation of an input/output system composed of only essential terms (i.e., excluding unnecessary terms) can be identified with high precision by using correct answer values that are provided as training data.

 

(3) It is confirmed that the proposed method is effective even in the presence of input variables that are unrelated to the input/output system. Furthermore, it is confirmed that “setting tolerance errors” and “expansion of the range for null modes” are necessary, when measurement error is included in the training data.