# Inverse Problems

Starting from noisy velocity images, unsteady three-dimensional blood flow and elastic vessel-wall motion can be reconstructed together without a prescribed inlet waveform or initial velocity. The solver alternates a fluid update, in which a physics-informed network fits the observations while enforcing the incompressible Navier–Stokes equations on the moving domain, with a geometry update, in which registration plus interface traction continuity and an elastic momentum residual constrain wall displacement. On a converging channel and a patient-like aorta, the PirateNets version reaches velocity relative errors of 0.0243 and 0.0372, below the 0.1066 and 0.1305 of a standard PINN of the same depth and width.

Starting from noisy velocity images, unsteady three-dimensional blood flow and elastic vessel-wall motion can be reconstructed together without a prescribed inlet waveform or initial velocity. The solver alternates a fluid update, in which a physics-informed network fits the observations while enforcing the incompressible Navier–Stokes equations on the moving domain, with a geometry update, in which registration plus interface traction continuity and an elastic momentum residual constrain wall displacement. On a converging channel and a patient-like aorta, the PirateNets version reaches velocity relative errors of 0.0243 and 0.0372, below the 0.1066 and 0.1305 of a standard PINN of the same depth and width.

Because an inverse map cannot be uniformly approximated by a continuous network near a discontinuity, the work learns a differentiable forward operator and its derivative and uses them as the surrogate inside Landweber iteration. On reaction-coefficient identification in an elliptic equation, four surrogates reconstruct both an in-distribution truth and one with sharp inclusions, but without preconditioning three of them semiconverge early and develop high-frequency spurious structure even on exact data; derivative supervision along fixed directions barely changes the best reconstruction.

Because an inverse map cannot be uniformly approximated by a continuous network near a discontinuity, the work learns a differentiable forward operator and its derivative and uses them as the surrogate inside Landweber iteration. On reaction-coefficient identification in an elliptic equation, four surrogates reconstruct both an in-distribution truth and one with sharp inclusions, but without preconditioning three of them semiconverge early and develop high-frequency spurious structure even on exact data; derivative supervision along fixed directions barely changes the best reconstruction.

In a heterogeneous medium, making a wave field take a prescribed shape at a set time means working backward to an initial pressure distribution, and previously every target and every medium required a fresh numerical inversion. JAX-WAVE trains a shared neural network to map the desired terminal field and a known wave-speed map directly to a bounded initial pressure, with training signal supplied by a differentiable finite-difference time-domain solver and no pre-optimized control labels. On an independent confirmation set, three trained models reach 25.24% error on unseen geometric targets and 4.77% on unseen reachable targets; within a 5 ms median online budget the learned map cuts mean error by 45% relative to accelerated numerical inversion on the heterogeneous benchmark.

In a heterogeneous medium, making a wave field take a prescribed shape at a set time means working backward to an initial pressure distribution, and previously every target and every medium required a fresh numerical inversion. JAX-WAVE trains a shared neural network to map the desired terminal field and a known wave-speed map directly to a bounded initial pressure, with training signal supplied by a differentiable finite-difference time-domain solver and no pre-optimized control labels. On an independent confirmation set, three trained models reach 25.24% error on unseen geometric targets and 4.77% on unseen reachable targets; within a 5 ms median online budget the learned map cuts mean error by 45% relative to accelerated numerical inversion on the heterogeneous benchmark.