A Physics-Agnostic Tensor-Train Framework for Discretized Parametric Solution Maps of Dynamical Systems

Wepresent a computational framework that builds the parametric solution map of a dynamical system as a family of tensor trains, one per state component, by cross-interpolation. A minimal contract separates the generic machinery from the specific dynamics: a physics module supplies the state derivative, the parameter axes with their ranges, and the index-to-state mapping, and nothing else changes between applications. Time is a tensor axis, so one integration fills a complete fiber for all state components; a bitwise-correct caching mechanism reduces the construction cost to one integration per distinct parameter point. We demonstrate the framework on a conservative cislunar problem, a dissipative atmospheric-flight problem, and a compartmental epidemic model, verified with instruments including an analytic vacuum limit, a nonlinear first integral, and a round-trip closure at least three orders below the reported errors. The construction attains a held-out accuracy of 3.6·10−7 at a compression factor of 8.9·105 on the first benchmark, of 6.6·10−4 at 7.8·102 on the second, and of 9.2·10−4 at 1.9·102 on the third. The rank rather than the grid is the binding constraint while the required rank stays below the grid cardinality, and the grid caps the attainable accuracy once it does not. The rank demand grows with the size of the uncertainty set and falls as it contracts: in the epidemic application, where every axis is a physical uncertainty, the rank falls to its minimum, while it plateaus in the other two applications. A comparison with polynomial chaos and Gaussian process surrogates on the same three families, under identical parameter boxes and error metrics, indicates that the difficulty of a parametric solution map is governed by the width of the parameter box direction by direction rather than by the number of directions.