Inverse prediction of upstream boundary conditions for nonlinear partial differential equations using artificial neural networks

The ill-posed inverse problem of estimating upstream boundary conditions from downstream measurements in convection-diffusion and Burgers' equations is addressed. Unlike the well-posed forward problem of predicting downstream data from upstream data, the inverse problem suffers from information loss due to diffusion and viscosity. Analytical investigation using the Volterra integral equation of the first kind shows that high-frequency components of the upstream signal are exponentially attenuated downstream, making direct reconstruction infeasible for neural networks. This analysis also provides criteria for selecting input resolution and temporal length by identifying the recoverable frequency range as a function of viscosity and sensor location. To address these difficulties, a composite network is proposed. A forward network modeling the downstream evolution of upstream data is coupled with an inverse-related network, forming a composite model that takes downstream measurements as input and outputs downstream data. The inverse-related component, trained only through the composite network without upstream boundary data, is optimized to infer an upstream condition that reproduces the downstream profile. Instead of directly learning the ill-posed inverse mapping, the framework avoids instability by training the forward and composite networks, which constitute a well-posed problem.