Generalized Neural Operator for Parametric and Boundary-Value Problems

Many physical systems are governed by Partial Differential Equations (PDEs), where dynamics are dictated by underlying physical parameters and boundary conditions. Modeling real-world phenomena requires a unified solver capable of generalizing across an entire family of PDE instances, yielding instant solutions for various parameters or boundary constraints without necessitating architectural redesign or retraining. However, existing approaches face a fundamental trilemma, failing to simultaneously achieve computational efficiency, mathematical well-posedness, and generalization. Traditional numerical solvers guarantee well-posedness but are computationally expensive and strictly instance-specific. Physics-Informed Neural Networks (PINNs) offer fast inference and physical fidelity, yet they remain instance-specific and require costly retraining for every new condition. Purely data-driven Neural Operators generalize across instances but treat physics implicitly. By attempting to infer parameters and boundaries from historical frames, they lead to ill-posed formulations. Recent large-scale foundational operators inherit this implicitness while scaling to billions of parameters, largely negating their computational advantage over optimized numerical baselines. To resolve this trilemma, we propose a Generalized Neural Operator that restores mathematical rigor through explicit physical conditioning. By formalizing the classical conditions for well-posedness within neural operators, our framework demonstrates the theoretical benefits of explicitly conditioning on PDE parameters and boundary conditions. To implement this synthesis without compromising computational speed, we introduce three novel architectural components: a parameter-gated mixture of kernels for efficient parameter generalization, a generalized boundary transfer operator that projects arbitrary boundary constraints into a unified latent Dirichlet representation, and a specialized training objective to ensure stability. Extensive experiments demonstrate that our theoretically grounded approach achieves superior generalization across heterogeneous physical regimes while maintaining inference efficiency comparable to conventional numerical baselines.