Active generative modeling: Reconstructing complex probability distributions via spectral analysis of the active Schrödinger equation

We propose a physics-based framework for generative modeling in active matter systems, bridging the gap between stochastic thermodynamics and score-based generative models. By exploiting the correspondence between the Fokker–Planck equation and the Schrödinger equation, we derive a method to reconstruct complex stationary distributions of Active Ornstein–Uhlenbeck Particles (AOUP). We utilize the Unified Colored Noise Approximation (UCNA) to define an effective “active potential” and solve the associated Schrödinger eigenvalue problem. This allows us to compute the exact time-dependent score function (the “Quantum Pilot”) via spectral decomposition, allowing the construction of an explicit, analytical score function without the need for data-driven training. We demonstrate this technique by reversing an algorithmic diffusion process for active particles in non-convex landscapes, including the Rosenbrock “Banana” potential and a double-well potential. Our results show that the spectral pilot successfully guides particles to recover specific initial microstates from a high-entropy state. Furthermore, we quantitatively analyze an effective “Barrier Transparency” mechanism within the UCNA description, showing that active dynamics reduce reconstruction error by orders of magnitude compared to passive Brownian diffusion in kinetically trapped landscapes.