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Title: Learning to Control PDEs with Differentiable Physics

Paper content:
Predicting outcomes and planning interactions with the physical world are long-standing goals for machine learning.
A variety of such tasks involves continuous physical systems, which can be described by partial differential equations (PDEs) with many degrees of freedom.
Existing methods that aim to control the dynamics of such systems are typically limited to relatively short time frames or a small number of interaction parameters.
We present a novel hierarchical predictor-corrector scheme which enables neural networks to learn to understand and control complex nonlinear physical systems over long time frames.
We propose to split the problem into two distinct tasks: planning and control.
To this end, we introduce a predictor network that plans optimal trajectories and a control network that infers the corresponding control parameters.
Both stages are trained end-to-end using a differentiable PDE solver.
We demonstrate that our method successfully develops an understanding of complex physical systems and learns to control them for tasks involving PDEs such as the incompressible Navier-Stokes equations.
Intelligent systems that operate in the physical world must be able to perceive, predict, and interact with physical phenomena (Battaglia et al., 2013) .
In this work, we consider physical systems that can be characterized by partial differential equations (PDEs).
PDEs constitute the most fundamental description of evolving systems and are used to describe every physical theory, from quantum mechanics and general relativity to turbulent flows (Courant & Hilbert, 1962; Smith, 1985) .
We aim to endow artificial intelligent agents with the ability to direct the evolution of such systems via continuous controls.
Such optimal control problems have typically been addressed via iterative optimization.
Differentiable solvers and the adjoint method enable efficient optimization of high-dimensional systems (Toussaint et al., 2018; de Avila Belbute-Peres et al., 2018; Schenck & Fox, 2018) .
However, direct optimization through gradient descent (single shooting) at test time is resource-intensive and may be difficult to deploy in interactive settings.
More advanced methods exist, such as multiple shooting and collocation, but they commonly rely on modeling assumptions that limit their applicability, and still require computationally intensive iterative optimization at test time.
Iterative optimization methods are expensive because they have to start optimizing from scratch and typically require a large number of iterations to reach an optimum.
In many real-world control problems, however, agents have to repeatedly make decisions in specialized environments, and reaction times are limited to a fraction of a second.
This motivates the use of data-driven models such as deep neural networks, which combine short inference times with the capacity to build an internal representation of the environment.
We present a novel deep learning approach that can learn to represent solution manifolds for a given physical environment, and is orders of magnitude faster than iterative optimization techniques.
The core of our method is a hierarchical predictor-corrector scheme that temporally divides the problem into easier subproblems.
This enables us to combine models specialized to different time scales in order to control long sequences of complex high-dimensional systems.
We train our models using a differentiable PDE solver that can provide the agent with feedback of how interactions at any point in time affect the outcome.
Our models learn to represent manifolds containing a large number of solutions, and can thereby avoid local minima that can trap classic optimization techniques.
We evaluate our method on a variety of control tasks in systems governed by advection-diffusion PDEs such as the Navier-Stokes equations.
We quantitatively evaluate the resulting sequences on how well they approximate the target state and how much force was exerted on the physical system.
Our method yields stable control for significantly longer time spans than alternative approaches.
We have demonstrated that deep learning models in conjunction with a differentiable physics solver can successfully predict the behavior of complex physical systems and learn to control them.
The in- troduction of a hierarchical predictor-corrector architecture allowed the model to learn to reconstruct long sequences by treating the physical behavior on different time scales separately.
We have shown that using a differentiable solver greatly benefits the quality of solutions since the networks can learn how their decisions will affect the future.
In our experiments, hierarchical inference schemes outperform traditional sequential agents because they can easily learn to plan ahead.
To model realistic environments, we have introduced observations to our pipeline which restrict the information available to the learning agent.
While the PDE solver still requires full state information to run the simulation, this restriction does not apply when the agent is deployed.
While we do not believe that learning approaches will replace iterative optimization, our method shows that it is possible to learn representations of solution manifolds for optimal control trajectories using data-driven approaches.
Fast inference is vital in time-critical applications and can also be used in conjunction with classical solvers to speed up convergence and ultimately produce better solutions.

assistant: text: We train a combination of neural networks to predict optimal trajectories for complex physical systems.