New Optimization Method Accelerates Convergence for Physics-Informed Neural Networks
Researchers have proposed N-RSAV, a new optimization algorithm that incorporates randomized low-rank Hessian approximation into the scalar auxiliary variable (SAV) framework to accelerate convergence on ill-conditioned problems. The method addresses a known weakness of existing RSAV-based optimizers, which rely only on first-order gradient information and converge slowly on problems like physics-informed neural networks (PINNs). By injecting second-order curvature information while preserving theoretical energy dissipation guarantees, N-RSAV offers a potentially significant practical improvement for a class of machine learning and scientific computing problems.
The preprint, submitted to arXiv on June 9, 2026, introduces the Nyström-enhanced relaxed scalar auxiliary variable method (N-RSAV), which augments the existing RSAV optimization framework with approximate second-order (Hessian) information derived from a randomized Nyström low-rank decomposition. Standard RSAV methods use only first-order gradient information and can converge slowly when the optimization landscape is ill-conditioned, a common issue in training physics-informed neural networks. To maintain the method's theoretical guarantees—specifically an unconditional modified energy dissipation law—the authors enforce positive semidefiniteness of the approximate Hessian operator via eigenvalue truncation. An adaptive reuse strategy is also introduced, allowing the algorithm to recycle previously computed Hessian approximations when the energy deviation is small, substantially reducing per-iteration computational overhead. Convergence is analyzed under the Polyak-Łojasiewicz (PL) condition for the general RSAV scheme, with additional convexity assumptions required for the full N-RSAV guarantees. Numerical experiments on convex quadratic problems and PINN training tasks demonstrate substantially faster convergence compared to conventional RSAV-based approaches, particularly on problems with effectively low-rank Hessian structure.
What's missing
The convergence guarantees for N-RSAV require both the PL condition and an additional convexity assumption, which may not hold in general deep learning settings beyond PINNs; the paper does not appear to benchmark against widely used second-order methods such as L-BFGS or K-FAC, making it difficult to assess competitiveness in a broader context. Scalability to very large-scale neural networks with dense or high-rank Hessians is not addressed.
What different sources said
- arXiv cs.LGCenter
Accelerating SAV-based optimization via randomized low-rank Hessian approximation
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