Verified residual-specific explicit derivative kernels for physics-informed learning and discretized PDE adjoints
Derivative computation is central to scientific computing, from space-time derivatives in physics-informed neural networks (PINNs) to residual Jacobian actions and discrete-adjoint operators in computational fluid dynamics (CFD). General-purpose automatic differentiation (AD) reduces implementation effort, but can incur substantial runtime and memory overhead for high-order residuals and complex discretized operators. Explicit derivative kernels can exploit problem-specific structure and provide efficient, controllable evaluations, but their use has been limited by derivation and implementation costs. This work revisits explicit differentiation (ED) as a residual-specific and verifiable route enabled by agent-assisted implementation and stringent numerical verification. For PINNs, we propose residual-specific partial-jet propagation, which makes the derivative-state closure of the target PDE residual explicit and realizes it through specialized layerwise kernels, rather than relying only on nested AD or a generic Taylor-mode transform. Relative to nested AD, the resulting ED kernels achieve floating-point-level agreement in residual and parameter-gradient evaluations and accelerate complete PINN training, often reaching 2-4x speedups while reducing peak GPU memory in most cases. For discretized PDE adjoints, we apply the same verification-driven strategy to a finite-volume CFD residual. The generated tangent-action and transpose-action kernels pass Taylor-remainder, inner-product, and reduced-gradient consistency checks, and are embedded into a GPU-resident discrete-adjoint workflow for freestream Mach-number and angle-of-attack inversion. These results suggest that verified explicit derivative kernels, supported by agent-assisted implementation, can serve as a practical, structure-aware complement to general-purpose AD for derivative-intensive scientific computing.