A built-in symmetry in passive networks makes nearly half of all learning targets impossible.

Scientists want to build mechanical networks that can learn tasks, like physical computers made of springs. Usually, people estimate how much these materials can learn just by counting their adjustable parts. But researchers realized this ignores a basic rule of physics called reciprocity, which forces passive, spring-like networks to behave symmetrically. Because of this symmetry, if the exact same p points are used to both send inputs and measure outputs, the network cannot learn every behavior. In fact, when inputs and outputs completely overlap, nearly half of the possible target behaviors become impossible to reach. This means any learning method gets stuck with a built-in minimum error that you can calculate before you even start training. Adding active external torque sources can restore the lost behaviors, but passive networks simply cannot learn tasks that break this physical symmetry rule.

Why it matters

Engineers building materials that learn from their environment face a hidden limit. This work shows that a basic physical symmetry cuts their design space in half, so some functions are physically impossible without adding external power.

Questions answered