Development of interpretable, data-driven plasticity models with symbolic regression

Highlights

• Symbolic regression is used to produce data-driven plasticity models in the form of yield equations.

• Tests are performed to verify a known plasticity model can be reproduced from response data with 95% robustness.

• A new, interpretable porous plasticity model is produced from response data of a representative volume element.

Abstract

In many applications, such as those which drive new material discovery, constitutive models are sought that have three characteristics: (1) the ability to be derived in automatic fashion with (2) high accuracy and (3) an interpretable nature. Traditionally developed models are usually interpretable but sacrifice development time and accuracy. Purely data-driven approaches are usually fast and accurate but lack interpretability. In the current work, a framework for the rapid development of interpretable, data-driven constitutive models is pursued. The approach is characterized by the use of symbolic regression on data generated with micromechanical finite element models. Symbolic regression is the search for equations of arbitrary functional form which match a given dataset. Specifically, an implicit symbolic regression technique is developed to identify a plastic yield potential from homogenized finite element response data. Through three controlled test cases of varying complexity, the approach is shown to successfully produce interpretable plasticity models. The controlled test cases are used to investigate the robustness and scalability of the method and provide reasonable recommendations for more complex applications. Finally, the recommendations are used in the application of the method to produce a porous plasticity model from data corresponding to a representative volume element of voids within a metal matrix.

Introduction

Though occasionally being criticized as less axiomatic than balance laws (e.g., mass, momentum), the development of constitutive models has helped enable the use of new materials and alloys. Specifically, elastoplastic constitutive models based on Tresca [1], Mises [2], and Hill [3] yield functions have stood as the bedrock for engineering analyses over the years. These three models, specifically focusing on plastic deformation, have reached this status because they posses two qualities: (1) they are accurate and tunable enough to be predictive, and (2) they are interpretable, allowing for good judgment in application and extension.

In contrast to the above constitutive models, which are rooted in physics and extended to fit observations, there has been a trend toward data-driven constitutive model development. Generally speaking these models embrace the notion that accuracy and predictive capability are more important then having a sound understanding of the physics involved. Neural network based constitutive models [4], [5], [6] are one example of this paradigm, wherein a neural network model is trained to predict a response (e.g., stress) as a function of load (e.g., deformation). The resulting neural net is often referred to as a black-box owing to its lack of physical insightfullness and interpretability. Another extreme of the data-driven paradigm is the recent work in data-driven computational mechanics which removes the idea of constitutive models altogether and use data directly for the constitutive relationship [7], [8], [9]. This method, by definition, is a black-box and offers no more insight into the constitutive relation than the underlying data itself.

There have also been examples of data-driven constitutive models that fall somewhere in the middle of the interpretablity spectrum. For instance, the work of Soare et al. [10], Soare and Barlat [11] that uses relatively opaque, higher-order polynomials as a plasticity model; however, the derivation and fitting of the coefficients preserves a well known aspect of plastic yield functions: convexity. Another example of partially interpretable constitutive models are based on hybrid modeling [e.g. 12], wherein the modeling error of a classical physics-based model is corrected using a black-box model.

Physics-informed machine learning is another paradigm that combines interpretable aspects of physics with black-box modeling [13], [14], [15]. In this variant of machine learning, physics are incorporated into the objective function for training of a black-box model; either in the form of a residual of a partial differential equation [14] or the weak form of the equivalent boundary value problem [15]. Because the approach in the current work also uses physics in training, it can be viewed as a physics-informed machine learning method; however, there are two key differences between the current work and other physics-informed machine learning works. Firstly, in contrast to hidden physics models [13] where coefficients in a fixed equation are found, a model is sought with arbitrary functional form. And secondly, the models that are produced in the current work are interpretable (in the form of relatively simple equations).

In many – if not most – cases the use of black-box models causes no issues, so long as the model is predictive. There are, however, advantages to interpretable plasticity models. Most importantly, interpretable constitutive models (1) can help drive knowledge discovery (2) may be used more confidently outside the range of their training data and (3) are usually easier to implement within current computational mechanics frameworks.

A common tool for the construction of interpretable, data-driven models is symbolic regression (SR). SR is a procedure for rapidly finding equations that fit a given dataset. Owing to recent advances in scalability, robustness, and efficiency [16], [17], [18], [19], [20], SR has reached maturity such that it can be used in many complex, real-wold applications. For example, Schmidt and Lipson [21] developed a method for performing implicit SR to discover conservation laws from experimental data. The focus of the current paper is the application of SR to the development of interpretable plasticity models. The two primary contributions of the current paper are (1) the formulation of the plasticity modeling problem as an implicit symbolic regression problem and (2) a framework for obtaining the relevant training data from micromechanical simulation.

The topic of using SR in the development of constitutive models has been investigated before. One early example is the work of Schoenauer et al. [22], which developed a three-dimensional hyperelastic model in terms of a strain energy function. The model had limited success for two reasons. First, inserting physical constraints upon the possible strain energy function proved difficult. Second, the presence of real-valued constants within the generated strain energy expressions caused difficulty in the SR optimization. The presence of real-valued constants have been an obstacle for the general SR community [23], [24]. Since the time of Schoenauer et al. [22], techniques have been developed to address the issue, such as restriction of the constants to coefficients of linearly combined SR terms (which allows for regression to obtain best-fit constants) [25], [26] as well as embedded nonlinear optimization of constants [27]. The latter is more computationally intensive but does not restrict the functional form obtained with SR.

For the most part, SR has been used in plasticity model development as a tool for modifying sub-components of a traditionally developed model. Examples include parametrically homogenized constitutive models, which use SR as a means of tying tunable parameters in a yield function to microstructural features [28], [29], [30]. Another example is the work of Versino et al. [31] who used SR in the creation of a flow stress model as part of a Mises ($J_2$) plasticity framework. They worked with relatively sparse experimentally-based datasets, because designing and performing experiments that could capture the needed data was costly and difficult. They were able to overcome the difficulty by supplementing their datasets with synthetic data points which represented their expert knowledge. Nonetheless, generation of significant quantities of relevant experimental data can be a challenge. For some applications there are high data-output experimental techniques [32], [33], [34] to feed data-hungry SR; however, in the current work an alternative approach is taken wherein data for SR is derived from micromechanical finite element (FE) simulations. The primary benefit is that simulation data is usually easier to generate than experimental data. Because the data is more plentiful, more complex relationships can be investigated. The underlying assumption of such an approach is that the micromechanical model accurately represents the material of interest.

In the current work, a framework is developed for using SR in the formulation of plastic yield functions based on the homogenized responses of micromechanical FE models. Details involved in the construction of the micromechanical FE models, along with the synthesis of their response data with SR, are covered in Section 2. The approach is then tested in three test cases (Sections 3 Test cases, 4 Test case results) illustrating its ability to generate interpretable and accurate plasticity models. Section 5 then discusses some of the key aspects that affect the accuracy and robustness of the method. Using the insights produced in Section 5, the method is applied to the problem of porous plasticity in Section 6. The major findings of this work are summarized in Section 7.