Chebyshev Center-Based Direction Selection for Multi-Objective Optimization and Training PINNs
For researchers training PINNs, this work provides a principled, interpretable, and unified framework for multi-objective optimization that outperforms existing methods, though it is domain-specific to PINNs.
The authors propose a Chebyshev-center-based method for selecting update directions in multi-objective optimization for training PINNs, which maximizes the minimum distance to dual cone facets. The method achieves convergence guarantees and strong empirical performance on benchmarks, unifying existing approaches under a single geometric principle.
Physics-informed neural networks (PINNs) are a promising approach for solving partial differential equations (PDEs). Their training, however, is often difficult because multiple loss terms induced by PDE residuals and boundary or initial conditions must be optimized simultaneously. To address this difficulty, existing approaches often construct update directions by explicitly enforcing particular desirable properties, such as scale robustness and simultaneous descent. While effective in many cases, such property-by-property designs can make it unclear which conditions are essential, what geometric principle determines the selected update direction, and how different methods are structurally related. In this work, we formulate update-direction selection for PINN training as a Chebyshev-center problem in the dual cone. The proposed formulation selects a normalized direction that maximizes the minimum distance to the cone facets. The resulting formulation admits an efficient dual problem in a much lower-dimensional space and yields a convergence guarantee in the nonconvex setting. It also recovers the key desirable properties targeted by existing approaches without imposing them separately; rather, they follow from the single geometric criterion underlying the formulation. This makes the selected direction interpretable through a single geometric rule and provides a unified basis for systematically comparing related direction-selection methods. Experiments on several PINN benchmarks further demonstrate strong empirical performance of the proposed method.