Curvature-Harmonized Metric Descent for Multi-Loss Optimization in PINNs

Training physics-informed neural networks (PINNs) requires jointly reducing multiple physics-based objectives despite gradient imbalance, directional conflicts, and ill-conditioned parameter-space geometry. Objective-aware methods can remain slow along ill-conditioned directions, whereas curvature-aware methods alone do not guarantee simultaneous descent across objectives. We propose **C**urvature-**H**armon**I**zed **ME**tric Descent (**CHIME**), which selects multi-objective directions in a shared curvature metric. **CHIME** maximizes the worst relative first-order reduction in residual norm under a step budget induced by this metric. The resulting direction lies in a curvature-harmonized hull, coupling a first-order non-conflict guarantee across objectives with curvature-aware preconditioning. A local linearized analysis shows that worst-objective progress and residual conditioning jointly control contraction of positive-eigenvalue residual modes. A practical SOAP-style realization estimates the shared metric and performs direction selection efficiently in objective space. Across four forward PINN benchmarks, **CHIME** reduces error by geometric mean factors of $5.85\times$ and $5.45\times$ relative to the best objective-aware and curvature-aware baseline for each benchmark, respectively. On the multi-task learning benchmark, it also achieves the best aggregate performance among the compared methods.