Many physical processes, such as the flow of air over a wing or the vibration of a membrane, are described in terms of fields constrained by partial differential equations (PDEs). Mechanisms that depend on controlling such systems may have to solve these equations, at least approximately, in real time. This kind of computation is currently very costly and requires large computational resources, which, for example, are not easily integrated into an airplane wing. An amorphous computer incorporating the sensors and actuators, integrated into the system to be measured and controlled, may be exactly what is required for this computationally demanding but highly local activity.
We can start to investigate this kind of computation by examining some elementary autonomous partial differential equations to see what it takes to compute their solutions by an intelligent material. Suppose we use each computational particle to represent a region of space (or space-time). If the computational particles actually know their positions accurately, then good methods are available to approximate the differential operators to high order, even though the neighbors are not related by a regular grid. But, unless they are the sources of physical measurements or the controls of physical actuators, it is not really necessary that the assumed positions of the particles are their actual positions. In fact, all that matters is that there is a consistent set of positions assigned to the processors (so that processors that think that they are near each other in the simulated space can communicate efficiently).
We shall show in the next section how one might arrange for particles to choose appropriate consistent positions for high-accuracy computations. However, even if the particles have no information about their positions, it still is possible to compute a low-accuracy solution to some PDEs, using an amorphous process that approximates the spatial locality of the formulation of the PDE with the local communication available to the particles.
One such PDE is Laplace's equation, , whose solutions are the harmonic functions. For any solution of Laplace's equation, each interior point is the average of the points around it. On a regular grid this yields a good approximation method that can be relaxed to get accurate solutions. However, even on a random grid, where there is no information about the distance or direction to each neighbor, our preliminary analysis and simulations shows that the process of replacing every point's value with the average of its communicating neighbors converges to an approximation to a solution of Laplace's equation. We need only set up appropriate boundary conditions to approximate a designated harmonic function.