Chebyshev Center-Based Direction Selection for Multi-Objective Optimization and Training PINNs

2026-05-11

AI Summary

This work addresses the challenge of coordinating multi-objective losses in the training of physics-informed neural networks (PINNs) by proposing a dual-cone Chebyshev center-based approach for selecting update directions. Specifically, the method determines a normalized gradient direction by maximizing the minimal distance to the cone boundary, thereby establishing a unified geometric criterion that replaces conventional per-task hyperparameter tuning. This formulation inherently ensures scale robustness and synchronous descent across objectives, while also providing convergence guarantees even in non-convex settings. Empirical evaluations on multiple PINN benchmark problems demonstrate that the proposed method significantly enhances training stability and solution accuracy, confirming its effectiveness and broad applicability.

Abstract

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.

Problem

• multi-objective optimization
• physics-informed neural networks
• update direction selection
• Chebyshev center
• loss balancing

Innovation

• Chebyshev center
• physics-informed neural networks
• multi-objective optimization
• dual cone
• direction selection