Combining Machine Learning and Simulation to a Hybrid Modelling Approach: Current and Future Directions

The main goal of machine learning is that a machine automatically learns a model that describes patterns in given data. The typical components of machine learning are illustrated in Fig. 2. In the first, main phase an inductive model is learned. Inductive means that the model is built by drawing conclusions from samples and is thus not guaranteed to depict causal relationships, but can instead identify hidden, previously unknown patterns, meaning that the model is usually not knowledge-based but rather data-based. This inductive model can finally be applied to new data in order to predict or infer a desired target variable.

The model generation phase can be roughly split into four sub-phases or respective components [1, 34]. Firstly, training data is prepared that depicts historical records of the investigated process or system. Secondly, a hypothesis set is defined in the form of a function class or network architecture that is assumed to map input features to the target variables. Thirdly, a learning algorithm tunes the parameters of the hypothesis set so that the performance of the mapping is maximized by using optimization algorithms like gradient descent and results in, fourthly, the final hypothesis, which is the desired inductive model. This model generation phase is often repeated in a loop-like manner by tuning hyper-parameters until a sufficient model performance is achieved.

The goal of a simulation is to predict the behaviour of a system or process for a particular situation. There are different types of simulations, ranging from cellular automata, over agent-based simulations, to equation-based simulations [9, 15, 36]. In the following we concentrate on the last type, which is based on mathematical models and is especially used in science and engineering. The first, required stage preceding the actual simulation is the identification of a deductive model, often in the form of differential equations. Deductive in this context means that the model describes causal relationships and can thus be called knowledge-based. Such models are often developed through extensive research, starting with a derivation, for example in theoretical physics, and continuing with plentiful experimental validations. Some recent research exists of proof-of-concepts for identifying models directly from data [8, 33].

The main phase of a simulation is the application of the identified model for a specific scenario, often called running a simulation. This phase can be described in four typical main components or sub-phases, which are, as illustrated in Fig. 3, the mathematical model, the input parameters, the numerical method, and finally the simulation result [36]. After the selection of a mathematical model, the input parameters that describe the specific scenario are defined in the second sub-phase. They can comprise general parameters such as the spatial domain or time of interest, as well as initial conditions quantifying the systems’ or processes’ initial status and boundary conditions defining the behaviour at domain borders. In the third sub-phase, a numerical method computes the solution of the given model observing the constraints resulting from the input parameters. Examples for numerical methods are finite differences, finite elements or finite volume methods for spatial discretization [36], or particle methods based on interaction forces [26]. These form the basis for an approximate solution, which is the final simulation result. This model application phase is often repeated in a loop-like manner, e.g., by tuning the discretization to achieve a desired approximation accuracy and stability of the solution.

Simulation offers an additional source of information for machine learning that goes beyond typically available data and that is rich of knowledge. This additional information can be integrated into the four components of machine learning as illustrated in Fig. 4. In the following, we will give an overview about these integration types by giving for each an illustrative example and refer for a more detailed discussion to [34].

Simulations are particularly useful for creating additional training data in a controlled environment. This is for example applied in autonomous driving, where simulations such as physics engines are employed to create photo-realistic traffic scenes, which can be used as synthetic training data for learning tasks like semantic segmentation [14], or for adversarial test generation [40]. As another example, in systems biology, simulations can be integrated in the training data of kernelized machine learning methods [13].

Moreover, simulations can be integrated into the hypothesis set, either directly as the solvers or through deduced, empirical functions that compactly describe the simulations results. These functions can be built into the architecture of a neural network, as shown for the application of finding an optimal design strategy for a warm forming process [20].

The integration of simulations into the learning algorithm can for example be realized by generative adversarial networks (GANs), which learn a prediction function that obeys constraints, which might be unknown but are implicitly given through a simulation [31].

Another important integration type is in the validation of the final hypothesis by simulations. An example for this comes from material discovery, where first a machine learning model suggests new compounds based on patterns in a data basis, and second the physical properties are computed and thus checked by a density functional theory simulation [17].

An approach that uses simulations along the whole machine learning pipeline is reinforcement learning (RL), when the model is learned in a simulated environment [2]. Studies under the keyword “sim-to-real” are often concerned with robots learning to grip or move unknown objects in simulations and usually require retraining in reality. An application for controlling the temperature of plasma follows the analogous approach, i.e., a training based on a software-physics model, where the learned RL model is then further adapted for use in reality [41].

Machine learning is often used in simulation with the intention to support the solution process or to detect patterns in the simulation data. With respect to our conceptual framework presented in Sect. 3, machine learning techniques can be used for the initial model, the input parameters, the numerical method, and the final simulation results, as illustrated in Fig. 4. In the following we will give an overview about the integration types. Again, we do not intend to cover the full spectrum of machine-learning assisted simulation, we rather want to illustrate its diverse approaches through representative examples.

A prominent integration type of machine learning techniques into simulation is the identification of simpler models, such as surrogate models [11, 12, 16, 26]. These are approximate and cheap to evaluate models that are particularly of interest when the solution of the original, more precise model is very time- or resource-consuming. The surrogate model can then be used to analyse the overall behaviour of the system in order to reveal scenarios that should be further investigated with the detailed original simulation model. Such surrogate models can be developed with machine-learning techniques either with data from real-world experiments, or with data from high-fidelity simulations. One application example is the optimization of process parameters using deep neural networks as surrogate models [27]. Kernel-based approaches are also commonly used as surrogate models for simulations, an example to improve the energetic efficiency of a gas transport network is shown in [10]. A well-established approach for surrogate modelling is model order reduction, for example with proper orthogonal decomposition, which is closely related to principal component analysis [5, 37].

Data assimilation, which includes the calibration of constitutive models and the estimation of system states, is another area where machine learning techniques enhance simulations. Data assimilation problems can be modelled using dynamic Bayesian networks with continuous physically interpretable state spaces where the evaluation of transition kernels and observation operators requires forward-simulation runs [29].

Machine learning techniques can also be used to study the parameter dependence of simulation results. For example, after an engineer executes a sequence of simulations, a machine learning model can detect different behavioral modes in the results and thus reduce the analysis effort during the engineering process [6]. This supports the selection of the parameter setting for the next simulation, for which active learning techniques can also be employed. For example, [39] studied it for selecting the molecules for which the internal energy shall be determined by computationally expensive quantum-mechanical calculations, as well as for determining a surrogate model for the fluid flow in a well-bore while drilling.

The integration of machine learning techniques into the numerical method can support to obtain the numerical solution. One approach is to exchange parts of the model that are resource-consuming to solve, with learned models that can be computed faster, for example with machine learning generated force fields in molecular dynamics simulations [26]. Another approach that is recently investigated are trainable solvers for partial differential equations that determine the complete solution through a neural network [28].

A further, very important integration type is the application of machine learning techniques on the simulation results in order to detect patterns, often motivated by the goal of scientific discovery. While there are plenty of application domains, two exemplary representatives are particle physics [3] and earth-sciences, for example with the use of convolutional neural networks for the detection of weather patterns on climate simulation data [30]. For further examples we refer to a survey about explainable machine learning for scientific discovery [32].