Learning Permutation-invariant Macroscopic Dynamics

Accurately modeling the macroscopic dynamics of high-dimensional microscopic systems is of broad interest across the sciences. Many data-driven approaches learn a low-dimensional latent state through an autoencoder trained for pointwise input reconstruction. These methods typically assume a fixed ordering of microscopic degrees of freedom in the input. However, in many settings, such as particle systems, the microscopic state is inherently unordered. This motivates an autoencoder framework that learns permutation-invariant latent representations. To this end, we adopt a permutation-invariant encoder and design the decoder to reconstruct the mass distribution centered at the observed points rather than per-sample reconstruction. We then jointly learn the macroscopic dynamics of the observables together with the latent states. We demonstrate the effectiveness and robustness of the proposed method across a range of microscopic settings, including learning the energy dynamics in interacting particle systems, predicting mixing dynamics in Lennard–Jones fluids, and modeling the stretching dynamics from video data of polymers moving in an elongational force field.

Many systems in nature are made up of a large number of small parts, such as particles in a fluid or beads in a polymer. Instead of tracking every small part in detail, scientists often want to predict larger-scale properties, such as how the temperature, energy, or mixing level of the system changes over time. This is challenging for machine learning because these small parts usually do not come with a meaningful order. For example, if we swap the labels of two identical particles, the physical system is still the same, but many machine learning models may treat it as different. We propose a method that avoids relying on such artificial ordering. Rather than arranging the particles into a fixed list, our model looks at the overall pattern formed by the particles and learns a compact summary of that pattern. It then uses this summary to predict how the larger-scale properties of the system will evolve. This also makes the method easier to apply when the number of particles changes. We test our approach on several examples, including interacting particles, mixing fluids, and stretching polymers. The results show that our method can predict important large-scale behavior without requiring scientists to manually choose an ordering for the small parts of the system.