Having finished force and mathematics → "Nature computes with that force"
This is the final cluster, "Force and Computation." In force and mathematics we saw that "nature makes the action stationary." Look at that through the eyes of computation, and something astonishing can be said ── nature is "computing" using force. Dip a complicated wire frame into soapy water and the film makes, in an instant, the surface of minimal area (the "minimal surface problem" that mathematicians struggle to solve, the film solves without thinking). Pull a spring network and let go, and it settles by itself into the balance of forces ── the solution of a system of simultaneous equations. A physical process that minimizes energy is, right there, "solving a problem." And the AI's gradient descent we saw in the final bonus was also a member of this large family.
Dip a wire frame into soapy water, and the film tries to reduce its area by surface tension (a force), settling into the surface of minimal area. That is exactly the answer to a hard mathematical problem (Plateau's problem): "what is the smallest-area film spanning a given boundary?" The film isn't solving an equation, yet just by obeying a force, it delivers the answer. A spring network is the same ── pull the nodes and let go, and it slackens by itself into the configuration where every spring's force balances = the solution of a system of simultaneous equations.
A physical system, obeying the force \(F=-\nabla(\text{energy})\) (Episode 6), rolls toward the valley of energy.
That valley (the balance of forces) is the answer to the problem you want to solve.
Soap film = minimal surface, spring network = the solution of simultaneous equations. To "solve" is to "minimize energy."
Why does force become computation? In Mathematics ① we saw that "nature chooses the path that makes the action (or energy) stationary." Translated into the language of computation ── write the problem you want to solve in the form "which configuration minimizes the energy?", and then just leave the physical system alone. The system falls into a valley obeying the force \(-\nabla E\), and where it stops is the answer. In the figure below, watch a spring network pulled into a bumpy shape slacken, obeying force, toward the valley of energy (= the balanced solution). Advance the steps and the ragged network settles into the smooth shape of the "answer."
This "solving with physics" has been used for a long time. An analog computer is wired so the voltages in an electrical circuit satisfy a differential equation; read the voltages once the circuit settles and you have the answer (letting a "system of forces" ── currents and voltages ── do the computing). Experiments finding shortest networks with soap films are also famous. And ── the AI learning (gradient descent) we saw in the final bonus is exactly this. It rolls down the landscape of the energy called loss, obeying the force \(-\nabla(\text{loss})\), and falls into a valley (the answer). Nature, analog computers, and AI are all one family: "solving by using force to minimize energy."
The conclusion of Force and Computation ①. Nature is computing by using force. A physical process carries out energy minimization, and its valley becomes the answer to a problem. Soap films give the minimal surface, spring networks give the solution of simultaneous equations, AI gives the loss-minimizing parameters ── all of them solve just by "obeying force to fall into the valley of energy." The force we peeled back to "force is a relation" in the main series was, at the same time as it moves the world, also a tool that makes the world solve problems.
"Solving with force" is not all-powerful. The system only falls into a nearby valley (a local minimum); it does not necessarily reach the deepest valley (the global minimum) (the same worry as AI learning). Relaxation takes time, and analog solutions have limits of precision and noise. So in practice, one uses thermal fluctuation (annealing) or tricks to escape valleys. "Physics solves without thinking" is a beautiful truth, but the honest fact is that it does not always give the best answer.
The figure is a schematic of a spring chain fixed at both ends (+ a uniform downward force) relaxed by the energy gradient, converging to the equilibrium shape (a discrete version of a parabola). It does not solve an actual soap film or a general system of simultaneous equations itself.
A soap film solves for the minimal surface, a spring network for simultaneous equations (the balance of forces), just by obeying force to fall into the valley of energy. To "solve" is to "minimize energy" ── the computational version of Mathematics ①'s "nature makes the action stationary." Analog computers (electrical circuits) and an AI's gradient descent (AI and Force) were all part of this same family. Force is not only what moves the world; it is also a tool for solving problems.
But the system only falls into a nearby valley (a local minimum) and is not always the best ── that's the honest line. Next time (Computation ②) is the flip side ── the laws of force are this simple, yet trying to solve them becomes uncomputable. The three-body problem and chaos. On to the wonder of how unpredictable complexity is born from simple force.
Print / save as PDF: ⌘+P (Ctrl+P on Windows). On screen, the slider lets you watch the spring network slacken into the shape of the "answer" by obeying force. "See the answer" opens each solution.