A cooperative game for automated learning of elasto-plasticity knowledge graphs and models with AI-guided experimentation

Abstract

We introduce a multi-agent meta-modeling game to generate data, knowledge, and models that make predictions on constitutive responses of elasto-plastic materials. We introduce a new concept from graph theory where a modeler agent is tasked with evaluating all the modeling options recast as a directed multigraph and find the optimal path that links the source of the directed graph (e.g. strain history) to the target (e.g. stress) measured by an objective function. Meanwhile, the data agent, which is tasked with generating data from real or virtual experiments (e.g. molecular dynamics, discrete element simulations), interacts with the modeling agent sequentially and uses reinforcement learning to design new experiments to optimize the prediction capacity. Consequently, this treatment enables us to emulate an idealized scientific collaboration as selections of the optimal choices in a decision tree search done automatically via deep reinforcement learning.

Appendix: Traction-separation example

In this appendix, we provide an additional simple example for traction-separation law to (1) demonstrate how to selection components from a combinations of existing hand-crafted and machine-generated operators/mappings, (2) provide one possible example to form the the directed labeled multi-graph from existing mappings in the plasticity literature and (3) show the inclusive nature of the multi-graph approach by demonstrating how to merge multiple sub-graph into a multi-graph.

Example 1

Traction-separation Law. Given a pre-defined objective function, assume that the only known theoretical traction-separation model incorporated in the labeled directed multigraph are the Tvergaard model (cf. [90]) and the Ortiz–Pandolfi model (cf. [61]). In addition, we also consider using a neural network that incorporates porosity to predict traction-separation relations. Define the labeled directed multi-graph that provides all the options available.

Fig. 17

The generation of directed multi-graph by expanding action space using previous models

First, we convert the traction-separation laws into directed graphs where the relative displacement vector is the input and the traction is the output. Notice that both Tvergaard [90] and Pandolfi et al. [61] are effective displacement models where an effective displacement $\overline{\Delta }$ is used as additional input to predict the traction. In [90],

$$\begin{aligned} T_{n}= & {} \frac{\overline{T}(\overline{\Delta })}{\overline{\Delta }}\frac{\Delta _{n}}{\delta _{n}}, \end{aligned}$$

(57)

$$\begin{aligned} T_{t}= & {} \frac{\overline{T}(\overline{\Delta })}{\overline{\Delta }} \alpha \frac{\Delta _{n}}{\delta _{t}} \end{aligned}$$

(58)

and the effective displacement and effective traction are scalars defined as,

$$\begin{aligned} \overline{\Delta }= & {} \sqrt{ (\Delta _{n}/ \delta _{n})^{2} + (\Delta _{t} / \delta _{t})^{2}}, \end{aligned}$$

(59)

$$\begin{aligned} \overline{T}(\overline{\Delta })= & {} \frac{27}{4} \sigma _{\max } \overline{\Delta }(1 - 2 \overline{\Delta } + \overline{\Delta }^{2}), \end{aligned}$$

(60)

where $\delta _{n}$ and $\delta _{t}$ are the characteristic length corresponding to the fracture energy and cohesive strength of the normal and tangential opening modes, $\alpha$ is a non-dimensional material parameter. As pointed out in [62], the traction-separation model in [61] can be expressed in the forms of Eqs. (57) and (58) with the alternative definition of effective displacement and traction separation law, i.e.,

$$\begin{aligned} \overline{\Delta }= & {} \tilde{\Delta } / \delta _{n} \, , \, \tilde{\Delta } = \sqrt{ \Delta _{n}^{2} + \beta ^{2} \Delta _{t}^{2}} \end{aligned}$$

(61)

$$\begin{aligned} \overline{T}(\overline{\Delta })= & {} k \overline{\Delta } + c \end{aligned}$$

(62)

where k is typically negative and c is the effective cohesive strength. Finally, we consider a neural network model in which the traction depends on the porosity $\phi ^{f}$ [16, 82, 92], i.e.,

$$\begin{aligned} T_{n}= & {} f^{\text {LSTM}}(\phi ^{f}, \Delta _{n}), \end{aligned}$$

(63)

$$\begin{aligned} T_{t}= & {} g^{\text {LSTM}}(\phi ^{f}, \Delta _{t}), \end{aligned}$$

(64)

where the exact expression of the function $f^{\text {LSTM}}$ and $g^{\text {LSTM}}$ are determined by adjusting the weight of the neurons in the recurrent neural network [36, 94]. Assuming that the solid constituent is incompressible, the porosity reads,

$$\begin{aligned} \phi ^{f} = \phi ^{f}_{o} (1 + \Delta _{n} \Delta _{t}) \end{aligned}$$

(65)

The multi-graph that combines all the possible choices of the three traction separation laws can therefore be defined by multi-graph statement with the following sets,

$$\begin{aligned} \mathbb {V}= & {} \{\Delta _{n}, \Delta _{t}, T_{n}, T_{t}, \overline{\Delta }, \overline{T}, \phi ^{f}\} \end{aligned}$$

(66)

$$\begin{aligned} \mathbb {E}= & {} \mathbb {E}_{1} \cup \mathbb {E}_{2} \cup \mathbb {E}_{3} \end{aligned}$$

(67)

$$\begin{aligned} \mathbb {E}_{1}= & {} \{ \Delta _{n} \rightarrow \overline{\Delta }, \Delta _{t} \rightarrow \overline{\Delta }, \Delta _{n} \rightarrow \phi ^{f}, \Delta _{t} \rightarrow \phi ^{f}, \Delta _{n} \nonumber \\&\quad \rightarrow T_{n}, \Delta _{t} \rightarrow T_{t} \} \end{aligned}$$

(68)

$$\begin{aligned} \mathbb {E}_{2}= & {} \{ \overline{\Delta } \rightarrow \overline{T}, \phi ^{f} \rightarrow T_{n}, \phi ^{f} \rightarrow T_{t}, \Delta _{n} \rightarrow T_{n} \} \end{aligned}$$

(69)

$$\begin{aligned} \mathbb {E}_{3}= & {} \{ \overline{T} \rightarrow T_{n}, \overline{T} \rightarrow T_{t} \} \end{aligned}$$

(70)

$$\begin{aligned} \mathbb {L_{V}}= & {} \{\text {normal disp., tan. disp., normal traction, tan.}, \nonumber \\&\quad \text {traction eff. disp., eff. traction, porosity} \} \end{aligned}$$

(71)

$$\begin{aligned} \mathbb {L_{E}}= & {} \{ \text {Eq.} (57), \text {Eq.} (58), \text {Eq.} (59), \text {Eq.} (60), \text {Eq.} (61), \text {Eq.} (62),\nonumber \\&\quad \text {Eq.} (63), \text {Eq.} (64), \text {Eq.} (65) \} \end{aligned}$$

(72)

Since $\varvec{n}_{\mathbb {V}}$ is a bijective mapping, the labeling of the vertices is trivial. The rest of the mappings, i.e. $\varvec{s}$, $\varvec{t}$ and $\varvec{n}_{\mathbb {E}}$ can be visualized in a labeled directed multigraph as shown in Fig. 17. Essentially, the process of creating the directed multigraph is to mathematically represent all the possible options modelers can have when they are tasked to create a constitutive model for a data set. $\square$