Automated design of complex dynamic systems

Abstract

Several fields of study are concerned with uniting the concept of computation with that of the design of physical systems. For example, a recent trend in robotics is to design robots in such a way that they require a minimal control effort. Another example is found in the domain of photonics, where recent efforts try to benefit directly from the complex nonlinear dynamics to achieve more efficient signal processing. The underlying goal of these and similar research efforts is to internalize a large part of the necessary computations within the physical system itself by exploiting its inherent non-linear dynamics. This, however, often requires the optimization of large numbers of system parameters, related to both the system's structure as well as its material properties. In addition, many of these parameters are subject to fabrication variability or to variations through time. In this paper we apply a machine learning algorithm to optimize physical dynamic systems. We show that such algorithms, which are normally applied on abstract computational entities, can be extended to the field of differential equations and used to optimize an associated set of parameters which determine their behavior. We show that machine learning training methodologies are highly useful in designing robust systems, and we provide a set of both simple and complex examples using models of physical dynamical systems. Interestingly, the derived optimization method is intimately related to direct collocation a method known in the field of optimal control. Our work suggests that the application domains of both machine learning and optimal control have a largely unexplored overlapping area which envelopes a novel design methodology of smart and highly complex physical systems.

Figure 1. Embedding a trajectory in an MSD-network.

A: Depiction of the initial condition of the MSD-system. The grey circles have fixed positions, the black circles are point masses that are non-fixed, and the red circle is the point mass of which we wish to control the trajectory. The connecting lines represent massless linear spring-dampers. B: Illustration of the trajectory of the selected node after optimisation. The full blue line is the actual trajectory, (including the trajectory after completing the pentagram) and the dashed line is the target. C: Comparison of the convergence speed of gradient descent and CMA-ES for the same initial parameter set.

Figure 2. MSD robots.

We have represented the springs that constitute the MSD robots with lines, where we visualized their modulation amplitude by making springs with large modulation amplitudes thicker and redder. The blue line represents the ground. A: Shape of the robot at the initialization of training. All springs have an equally large modulation amplitude. B: Snapshot of an example robot after finishing training. Only 4 of its springs still have a non-negligible modulation amplitude, and they provide virtually all locomotive power.

Figure 3. Illustration of the magnetic beam focuser.

A: Particle distribution in a cross-section of the focal plane, shown for 50,000 particles, both for a set of two ideal quadrupoles and our configuration (CT-RTRL results). The cost function, the root mean squared distance (RMSD) of the particles w.r.t. the focal point as they pass through the focal plane, is shown underneath the panels. On the right we show the scale compared to the original beam width (red circle). Note that the spread of the particles for the quadrupoles is largely due to a relatively large spread in particle velocities. B: Illustration of the particle beam envelope (light green) and the spatial configuration of magnets (blue red cones), indicating position, direction and magnitude. C: Cut-through illustrations of the particle beam envelope and the lateral magnetic field at different positions throughout the beam focuser.

Figure 4. Results for optimised SOA networks.

A: Schematic representation of the SOA-network. The black lines are photonic waveguides. The circles are the SOAs, with the arrows indicating in which direction the light passes through. Each SOA receives an external input signal, and at its output, a channel goes to the output (represented by the dotted lines). These channels are optimally combined optically and the output signal is the optical power of this combination. B: Illustration of the photonic D flip-flop. The top two time traces show the two input channels, the clock (set) signal and the data signal. The red time trace is the desired output power of the SOA network, and the blue one is a superposition of the measured output power for 10 different instances, each with different noise and internal phase variations. C: Illustration of the photonic 5-bit one-hot detector. The black time trace is the input signal, a random bit stream, and the red time trace is the desired output power of the SOA network. The blue one is a superposition of the measured output power for 10 different instances, each with different noise and internal phase variations.