Reconciling approximate ordinary differential equations and noisy data using theory of functional connections

Abstract The application of theoretical models to real-world problems is often accompanied by uncertainty regarding the validity of the governing differential equations and the experimental data. This paper addresses the challenging problem of solving ordinary differential equations when the boundary conditions are unknown and require estimation from noisy experimental observations. The proposed method employs a weighted least-squares approach to estimate the solution and the associated unknown boundary conditions simultaneously. A key feature of the approach is the introduction of a parameter that controls the trade-off between the reliability of the governing equation and the fidelity of the experimental data. The problem is formulated and solved using the theory of functional connections, which enables the construction of solutions that inherently satisfy the differential equation under the specified boundary conditions while optimizing agreement with noisy data. The effectiveness of the proposed approach is demonstrated through a series of numerical examples and two applications to real-world problems involving the estimation of soil moisture—a variable relevant to weather forecasting and water resource management—and the modelling of biochemical reactions. These applications demonstrate the method's robustness and effectiveness in modelling complex biological and environmental systems.