1 Motivation and objectives
5 to 0. This is the score by which AlphaStar–a computer program developed by the artificial intelligence (AI) company DeepMind–beat a top professional player in Starcraft II, one of the most complex video games to date (Vinyals et al., 2019). This accomplishment tops the list of sophisticated human tasks at which AI is now performing at a human or superhuman level (Perrault et al., 2019), and enlarges the previous body of achievements in game playing (Silver et al., 2017, 2018) and in other tasks, e.g. medical diagnoses (Esteva et al., 2017; Raumviboonsuk et al., 2019; Nagpal et al., 2019).
Our focus is on computational modeling, which is of paramount importance to the investigation of many natural and industrial processes. Reduced-order models alleviate the computational intractability frequently associated with solving the exact mathematical description of the full-scale dynamics of complex systems. The motivation of this study is to leverage recent breakthroughs in AI research to unlock novel solutions to important scientific problems encountered in computational science. Following the post-game confession of the human player defeated by AlphaStar: “The agent demonstrated strategies I hadn’t thought of before, which means there may still be new ways of playing the game that we haven’t fully explored yet" (Vinyals et al., 2019), we inquire current AI potential to discover scientifically rooted models that are free of human bias for computational physics applications.
Traditional reduced-order modeling approaches are rooted in physical principles, mathematics, and phenomenological understanding, all of which are important contributors to the human interpretability of the models. However these strategies can be limited by the difficulty of formalizing an accurate reduced-order model, even when the exact governing equations are known. A notorious example in fluid mechanics is turbulence: the flow is accurately described in detail by the Navier-Stokes equations; however, closed equations for the large-scale quantities remain unknown, despite their expected simpler dynamics. The work by Jimenez (2020) discusses in more extent the role of computers in the development of turbulence theories and models.
To address the human intelligence limitations in discovering reduced-order models, we propose to supplement human thinking with artificial intelligence. Our work shares the goal of a growing body of literature on data-driven discovery of differential equations, which aims at substituting or combining traditional modeling strategies with the application of modern machine learning algorithms to experimental or high-fidelity simulation data (see (Brunton et al., 2019)
for a review on machine learning and fluid mechanics). Early work used symbolic regression and evolutionary algorithms to determine the dynamical model that best describes experimental data(Bongard & Lipson, 2007; Schmidt & Lipson, 2009). An alternative, less costly approach consists of selecting candidate terms in a predefined dictionary of simple functions using sparse regression (Brunton et al., 2016; Rudy et al., 2017; Schaeffer, 2017; Wu & Zhang, 2019). Raissi et al. (2017) and Raissi & Karniadakis (2018)
use Gaussian processes to learn unknown scalar parameters in an otherwise known partial differential equation. A recent method exploits the connection between neural networks and differential equations(Chen et al., 2018) to simultaneously learn the hidden form of an equation and its numerical discretization (Long et al., 2018, 2019). Atkinson et al. (2019)
use genetic programming to learn free-form differential equations. The closest work to ours in spirit is that ofLee et al. (2019), in which the authors identify coarse-scale dynamics from microscopic observations by combining Gaussian processes, artificial neural networks, and diffusion maps. It is also worth mentioning the distinct yet complementary work by Bar-Sinai et al. (2019)
, which deals with the use of supervised learning to learn discretization schemes.
Our three-pronged strategy consists of learning (i) models expressed in analytical form, (ii) which are evaluated a posteriori, and (iii) using exclusively integral quantities from the reference solution as prior knowledge. In point (i), we pursue interpretable models expressed symbolically as opposed to black-box neural networks, the latter only being used during learning to efficiently parameterize the large search space of possible models. In point (ii), learned models are dynamically evaluated a posteriori in the computational solver instead of based on a priori information from preprocessed high-fidelity data, thereby accounting for the specificity of the solver at hand such as its numerics. Finally in point (iii), the exploration of new models is solely guided by predefined integral quantities, e.g., averaged quantities of engineering interest in Reynolds-averaged or large-eddy simulations (LES). This also enables the assimilation of sparse data from experimental measurements, which usually provide an averaged large-scale description of the system rather than a detailed small-scale description. We use a coupled deep reinforcement learning framework and computational solver to concurrently achieve these objectives. The combination of reinforcement learning with objectives (i), (ii) and (iii) differentiate our work from previous modeling attempts based on machine learning.
The rest of this brief is organized as follows. In Section 2, we provide a high-level description of the model discovery framework with reinforcement learning. In Section 3, the method is detailed for the application of discovering missing terms in differential equations. An elementary instantiation of the method is described that discovers missing terms in the Burgers’ equation. Concluding remarks are offered in Section 4.
2 Model Discovery with Reinforcement Learning (MDRL)
2.1 Offload human thinking by machine learning
The purpose of modeling is to devise a computational strategy to achieve a prescribed engineering or physical goal, as depicted in Figure 1. For example, we may search a subgrid-scale (SGS) model for LES (strategy) that can accurately predict the average mass flow in a pipe (goal). While it is natural that determining the goal is a completely human oriented task, as we aim to solve a problem of our own interest, there is no obvious reason why this should be the case for the strategy.
Traditionally, as shown in Figure 1(a), finding a modeling strategy heavily relies on human thinking, which encompasses phenomenological understanding, physical insights, and mathematical derivations, among others. Deriving a model is typically an iterative process that involves testing and refining ideas sequentially. Although strategies devised by human intelligence alone have shown success in the past, many earlier models can be substantially improved, and many remain to be found. Current modeling strategies may be hindered by the limits of human cognition, for example by a researcher’s preconceived ideas and biases. In a sense, we have a vast knowledge about traditional models but very limited idea about what we really seek: breakthrough models. This often constrains us to exploit prior knowledge for convenience more than we explore innovative ideas. Following the example in the previous paragraph, we can constrain the functional form of the SGS model in LES to be an eddy viscosity model and focus on the eddy viscosity parameter alone. This approach certainly facilitates the strategy search as the space of all possible models is often too large for researchers to exhaustively explore all of them. However, it is perhaps at the expense of constraining the final strategy to a suboptimal solution from the beginning of the modeling process.
Ideally, we would like to offload all human thinking of strategies to artificial intelligence to efficiently explore the phase space of models with minimum human bias, as sketched in Figure 1(b). Following the previous example, our goal is to predict the average mass flow, but whether we use LES or another computational approach, broadly defined, needs not be decided by a human, but instead by an artificial intelligent agent. To automate the full strategy process pertains to the bigger quest of general artificial intelligence (Goertzel & Pennachin, 2007)
. The aim of the present preliminary study is more modest. Yet, it is our premise that current machine learning tools already enable to automate a significant portion of the modeling strategy search. In the previous example where we seek a model to estimate the pipe mass flow, we might constrain the strategy to employ a LES framework, then use the artificial intelligent agent to specifically find the SGS model.
2.2 Hybrid human/machine method based on reinforcement learning
We pursue a method in which a reinforcement learning (RL) agent is used to partially replace human thinking in devising strategies. RL emulates how humans naturally learn and, similarly, how scientists iterate during the modeling process. In particular, we draw inspiration from the recent success of RL in achieving superhuman performance across a number of tasks such as controlling robots (OpenAI et al., 2018) or playing complicated strategy games (Silver et al., 2017, 2018; Vinyals et al., 2019).
RL consists of training a software agent to take actions that maximize a predefined notion of cumulative reward in a given environment (Sutton & Barto, 2018): the RL agent selects an action based on its current policy and state, and sends the action to the environment. After the action is taken, the RL agent receives a reward from the environment depending on the success in achieving a prescribed goal. The agent updates the policy parameters based on the taken action and the reward received. This process is continuously repeated during training, allowing the agent to learn an optimal policy for the given environment and reward signal.
The MDRL methodology proposed here relies on a tailored RL framework that mimics the scientific modeling process, as illustrated in Figure 2. MDRL consists of training a software scientist (agent with a given policy) to iteratively search for the optimal strategy or model (action) that maximizes the prescribed goals (reward) obtained by using the model in a computational solver (environment). As the state typically encountered in RL does not play a role here, MDRL may be considered an example of the multi-armed bandit problem (Sutton & Barto, 2018).
3 Application of MDRL for analytical model discovery
Hereafter, we narrow the term model to refer to mathematical expressions. Instead of hand-designing new models from scratch, we design a RL agent that searches for analytical formulas among the space of known primitive functions. In Section 3.1, we describe how MDRL automates the process of discovering models in analytical form. An elementary instantiation of the method is presented in Section 3.2, where MDRL is utilized to efficiently discover missing terms in the Burgers’ equation.
3.1 Description of the method
The workflow of MDRL for analytical model discovery is illustrated in Figure 3. The steps are summarized as follows:
A random model generator (RMG) is the agent that outputs mathematical expressions (actions) with a given probability distribution, similarly to a sequence of numbers generated by a random number generator. The RMG comprises a computational graph to encode mathematical formulas of arbitrary complexity in a particular domain-specific language (DSL). The probabilities defining the RMG are parameterized as a neural network, which represents the policy of the RL agent.
A set of models is sampled from the RMG and decoded into their corresponding mathematical expressions ( in Figure 3).
The sampled models are evaluated in the real environment using a computational solver.
The reward signal is calculated based on the prescribed goals and used to optimize the parameters (probabilities) of the RMG via a particular RL algorithm.
The process described in Steps 1 to 4 is repeated until a convergence criterion is satisfied.
The framework provided by MDRL has the unique feature of allowing the researcher to focus on the intellectually relevant properties of the model without the burden of specifying a predetermined analytical form. The final outcome is an equation modeling the phenomena of interest that can be inspected and interpreted, unlike other widespread approaches based on neural networks. Our method generates a mathematical equation, but could be similarly applied to generate symbolic expressions for numerical discretizations.
We discuss next some of the components of MDRL for analytical model discovery.
Domain-specific language: A Domain-specific language (DSL) is used to represent mathematical expressions in a form that is compatible with the reinforcement learning task. The purpose of the DSL is to map models to actions in the RL framework. Mathematical expressions can be generally represented as computational graphs, where the nodes correspond to operations or variables. Compatible representations of these computational graphs can employ sequence-like or tree-like structures (Bello et al., 2017; Luo & Liu, 2018; Lample & Charton, 2019), with varying degree of nesting depending on the requested complexity of the sought-after mathematical expression. It is important to remark that the search space is not restricted to elementary physical processes as the DSL allows for complex combinations of elementary operations to build new ones, similarly as to how with a few letters one can write a large number of words. The choice of the DSL plays an important role as it conditions the search space and therefore the learning process.
Random model generator as a neural network: The key insight that enables the RL agent to efficiently learn a model in the vast search space is to implicitly parameterize the latter using a neural network. The RMG (agent) employs a stochastic policy network to decide what models (actions) to sample. Actions are sampled according to a multidimensional probability distribution , where is the neural network policy and its parameters. The objective is to train the RMG to output models with increasing accuracy by learning a probability distribution function that maximizes the probability of sampling accurate models. The RL training objective can be formulated as maximizing
where is the averaged accuracy (reward) of an action (), sampled according to a probability distribution . The objective is to maximize the expected performance obtained when sampling a model from the probability distribution outputted by the policy. This is achieved by gradient ascending on the performance objective , using policy gradient algorithms for example.
A distributed training scheme can be employed to speed up the training of the RMG. As the model evaluation is generally the most time-consuming bottleneck, it is desirable to evaluate generated models in parallel on distributed CPUs. In this manner, at each iteration, the RMG samples a batch of models that are all run simultaneously and later combined to update the probabilities of the RMG.
3.2 Example: discovering missing terms in the Burgers’ equation
We apply the scheme proposed in Section 3.1 to discover the analytic form of missing terms in a partial differential equation. We choose the Burgers’ equation as representative of the key features of a simple fluid system,
where denotes velocity, and are the spatial and temporal coordinates, respectively, and is the viscosity. We simulate Eq. (2) using as a initial condition a rectangular signal. The equation is integrated from to
using a fourth-order Runge-Kutta scheme for time-stepping and a fourth-order central finite difference scheme for approximating the spatial derivatives with 1000 points uniformly distributed in. The velocity profile is plotted at various time instants in Figure 4.
The problem is formulated by re-arranging Eq. (2) as
where is an unknown mathematical expression that we aim to discover using the MDRL framework proposed in Figure 3. For each model sampled by the RMG, we compute the associated reward as
where is the L-norm, is the exact solution at , is the solution for the -th model at , is the number of terms in (for example for ), and . The first term in evaluates the accuracy of the model, whereas the second term penalizes the model based on its complexity (number of terms). In each iteration of the RL process, the RMG samples models, which results in the total reward . The reward is then used to improve the RMG.
We discuss next the implementation of the RMG and the neural network architecture for the RL agent used in this particular example. Both the proper design of the RMG agent along with the choice of the optimization method plays a major role in the success of the proposed methodology. The specific choices made here have shown acceptable performance, but we do not imply that these are optimal or extensible to other cases.
For the computational graph representation of , we follow a DSL methodology similar to that used by Bello et al. (2017). Models are formed by combining operands, unary functions, and binary functions. Each group is composed of a few elementary elements:
Operands: , , , and integers from 1 to 100 and the reciprocals .
Unary functions: (identity), (sign flip), , , , , and (differentiation).
Binary functions: (addition), (subtraction), (multiplication), and (division).
Formulas with an arbitrary number of terms are generated following a recursive scheme similar to the one depicted in Figure 3. The details of the specific DSL strategy adopted here can be cumbersome. They are not emphasized as they are merely designed for the particular showcase discussed in this example. Further details regarding the unique and efficient representation of analytical formulas using computational graphs can be found in Bello et al. (2017), Luo & Liu (2018), and Lample & Charton (2019).
We use Deep Deterministic Policy Gradients (DDPG) (Lillicrap et al., 2016)
as the network architecture for the RL agent. The DDPG actor-critic algorithm is well suited for problems with continuous action spaces, which we assimilate to the probabilities required for the RMG in the current setting. The DDPG is implemented using MATLAB (R2019a, The MathWorks Inc.) with default parameters. The actor (RMG agent) is a multilayer perceptron with two blocks, each with 5 fully connected hidden layers. Rectified linear units and sigmoid activations are used in the first and second blocks, respectively. The critic neural network is a multilayer perceptron with 5 fully connected hidden layers with rectified linear units as activation functions. Each layer contains roughly one hundred neurons.
The probability of finding the exact solution, , during the learning process is shown in Figure 5. The result is obtained by performing 100 independent learning processes starting from scratch (RMG with uniform probability distribution). After approximately 220 iterations, the probability of discovering the exact solution is 99%.
Our results can be compared with a random search approach, i.e. attempting to discover the model according to a fixed RMG with uniform probability distribution. The latter typically requires more than iterations to find the exact solution. Note that this random search approach was possible in the present example, but for cases with a vastly larger phase space, as in real-world applications, the random search becomes intractable and is likely to fail in finding an accurate model. Hence, the results suggest that MDRL is successful in detecting useful patterns in mathematical expressions and thereby speeds up the search process.
In this preliminary work, we discuss a paradigm for discovering reduced-order computational models assisted by deep reinforcement learning. Our main premise is that state-of-the-art artificial intelligence algorithms enable to offload a significant portion of human thinking during the modeling process, in a manner that differs from the conventional approaches followed up to date.
We have first divided the modeling process into two broad tasks: defining goals and searching for strategies. Goals essentially summarize the problem we pursue to solve, whereas strategies denote practical solutions employed to achieve these goals. Within the framework discussed here, we have argued that defining goals ought to remain in the realm of human thinking, as we aim to solve a problem of our interest. In contrast, strategies are mere practical means to achieve these goals. Thus we have advocated for a flexible methodology that intelligently search for optimal strategies while reducing scientists’ biases during the search process. We deemed the latter necessary to prevent misleading preconceptions from hindering our potential to discover ground-breaking models.
In this brief, we presented the main ideas behind a hybrid human/AI method for model discovery by reinforcement learning (MDRL). The approach consists of learning models that are evaluated a posteriori using exclusively integral quantities from the reference solution as prior knowledge. This allows the use of a wide source of averaged/large-scale computational and experimental data. The workflow is as follows. The scientist sets the goals, which defines the reward for the reinforcement learning agent. An intelligent agent devises a strategy that ultimately results in a model candidate. The model is evaluated in the real environment. The resulting performance, as measured by the reward, is fed back into the reinforcement learning agent that learns from that experience. The process is repeated until the model proposed by the agent meets the prescribed accuracy requirements. In the long term, it would be desirable to offload all human thinking of strategies to artificial intelligence. In this preliminary work, only a portion of the strategy is discovered by machine learning.
We have detailed the scheme above by combining MDRL with computational graphs to learn analytical mathematical expressions. The approach comprises two components: a random model generator (agent) that generates symbolic expressions (models) according to a parameterized probability distribution, and a reinforcement learning algorithm that updates the parameters of the model generator. At each iteration, the agent learns to output better models by updating its parameters based on the error incurred by the tested models. As an example, we have applied MDRL to discovering missing terms in the Burgers’ equation. This simple yet meaningful example shows how our approach retrieves the exact analytical missing term in the equation and outperforms a blind random search by several orders of magnitude in terms of computational cost.
Although the main motivating examples in this brief pertain to the field of fluid mechanics, the MDRL method can be applied to devising computational models or discovering theories in other engineering and scientific disciplines. Finally, the present brief should be understood as a statement of concepts and ideas, rather than a collection of best practices regarding the particular implementations (computational graphs, neural network architectures, etc.) of the MDRL for general problems. Refinement and further assessment of the method are currently investigated and will be discussed in future work.
A.L.-D. acknowledges the support of NASA under grant No. NNX15AU93A and of ONR under grant No. N00014-16-S-BA10. We thank Irwan Bello for useful discussions. We also thank Jane Bae for her comments on the brief.
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