A handful of sand falls onto a table: each grain rolls down until it rests in a hollow. That's how a potential works: the table's surface tells the grains where to go. But in a river with whirlpools, there's no single basin everything flows into. Water swirls, returns, and meanders. For such systems, the usual map of 'pits and hills' doesn't apply.

The authors of the new study propose a universal tool — a generalized Lyapunov function. It's simply a number at each point that never increases along the system's actual path. If you can find such a function, you can see the hidden landscape: where motion slows down, where it gets stuck, and where it's carried onto a repeating orbit. Previously, such functions were built for specific cases. For instance, Freidlin and Wentzell introduced a quasipotential for systems with weak noise, and Ao proposed his own potential for biological networks. But there was no general recipe.

The researchers used a neural network that requires no data — only the system's equations themselves. The loss function (what the network minimizes during training) was embedded with two conditions: violation of the Lyapunov inequality and a weak compatibility condition with noise. The first ensures the function doesn't grow; the second adds robustness to random perturbations. The network was trained on linear systems, the Hopf bifurcation (a model for the onset of oscillations), the Van der Pol oscillator (a pendulum with nonlinear friction), and a three-dimensional flow called the 'Hopf ring'—two coupled limit cycles shaped like interlocking loops.

The result is striking: the learned function spontaneously revealed the system's structure. Stable points and closed orbits became basins or plateaus, while unstable regions became ridges. In the case of interlocking cycles, the landscape resembled two intertwined grooves along which the trajectory slides without violating the overall descent. This is like a mountain switchback: the road winds, but the altitude still decreases.