Aster: Learning Tensor-Network Structure for PDE Solving with a Differentiable Quantum–Classical Model

Tensor networks offer compact representations of high-dimensional fields and are increasingly used for solving partial differential equations (PDEs) at resolutions where conventional discretisations become computationally prohibitive. However, most tensor-network PDE methods rely on fixed structures chosen in advance, while existing structure-search methods primarily optimise compression of individual fields rather than PDE dynamics. Although quantum circuits provide flexible trainable samplers, their use for tensor-network structure discovery is further limited by the lack of efficient mechanisms for generating only valid network structures. We introduce Aster, a hybrid quantum–classical model that jointly learns tensor-network structures and PDE dynamics. A trainable quantum sampler generates valid network topologies without rejection and integrates them into a differentiable tensor-network solver. Aster${}^{+}$ further incorporates rank adaptation into the discovered network structure. Across three evaluation settings, Aster${}^{+}$ matches the accuracy of spectral Runge–Kutta on six of seven benchmark PDEs while using 2.9–10$\times$ fewer stored scalars on five. Aster-Solver achieves the lowest error at every evaluated resolution on eight of twelve two-dimensional PDEs against state-of-the-art neural PDE solvers, while using substantially fewer parameters. As a structure-search method, Aster${}^{+}$ also achieves lower storage at matched accuracy than existing approaches at the three highest resolutions.