Deep learning excels in error-tolerant applications with abundant data, but struggles in scientific settings where accuracy is critical, such as solving physics-based equations with limited observations. The core challenge lies in non-convex optimization, where traditional training lacks guarantees of reliability. This project develops a rigorous framework to control optimization accuracy by aligning iterative updates with mathematically sound “ideal descent paths” derived from the underlying physics. By dynamically adapting model complexity and certifying each step’s accuracy, we aim to overcome the unpredictability of non-convex optimization, enabling trustworthy artificial intelligence (AI) for high-stakes applications in engineering, medicine, and beyond. This work provides foundational tools to ensure AI-driven scientific predictions are both accurate and actionable. This project addresses the fundamental challenge of uncertain optimization success in physics-informed deep learning. Non-convex objective landscapes severely impede on accuracy control when dealing with error-sensitive problems, especially those involving partial differential equations (PDEs). The proposed approach establishes a mathematically grounded framework that enforces optimization accuracy control through residual-based loss functions that are “variationally correct”. This means that the loss is always proportional to the current approximation error with respect to model-compliant norms derived