Closure-Guided AI for Research-Level PDE Analysis

English abstract Research-level PDE analysis requires more than locally correct derivations: estimates must jointly control the derivatives, dissipative directions, parameters, and time scales required by the final theorem. We propose a closure-guided AI research method for nonlinear dissipative PDEs. Starting from a fixed research objective and its established foundations, the system uses the equation’s coupling and dissipation structure to organize variable choices, decompositions, multipliers, and energy estimates. Candidate estimates retain their mathematical objects and parameter dependencies in an explicit research state. For tractable composition subproblems, checkable constraints identify missing control; unresolved analytic questions remain explicit obligations. These obstructions guide the next mathematical operation rather than merely triggering another unrestricted proof attempt. Goal-preserving repair is distinguished from changing the theorem’s assumptions. We design budget-matched comparisons, operation-level interventions, and held-out parent problems in fluid and kinetic analysis to measure valid completion of the original objective, unsupported claims, and research cost. A Boltzmann contact-wave investigation supplies a demanding longitudinal case, without treating its unresolved analytic inputs as established results. The central question is whether PDE structure can become an operational constraint on proof search, enabling AI systems to construct arguments whose estimates support a complete research conclusion.