Accelerating SAV-based optimization via randomized low-rank Hessian approximation

Physics-informed neural networks (PINNs) have emerged as a promising route to solve partial differential equations, yet they have struggled to reach the precision of classical solvers. The obstacle is increasingly understood to be one of optimisation, owing to the severely ill-conditioned loss landscape. We present $\textbf{DSGNAR}$: Doubly-Sketched Gauss-Newton with Adaptive Ratio, a scalable second-order optimisation framework that confronts this ill-conditioning and, in doing so, obtains unprecedented accuracy and speed. $\textbf{DSGNAR}$ couples a doubly-sketched Gauss-Newton model with a novel strategy that carefully controls both regularisation and step length. Across a suite of problems spanning nonlinear, chaotic, multi-scale, high-dimensional, and Navier-Stokes, the framework greatly improves on the state of the art: able to attain relative $\ell_2$ errors as low as $3\times10^{-16}$ in double precision, improve contemporary results by five orders of magnitude on the canonical Burgers' equation, and as much as eight orders on a high-dimensional Poisson problem, while remaining markedly faster. We further show that, in single precision, solutions at the limit of round-off error can be obtained very quickly: Burgers' equation to $\ell_2^{\text{rel}} = 4.75 \times 10^{-7}$ in under ten seconds. The framework is also robust to the choice of architecture, arithmetic precision, and initial hyperparameters. The code is available at https://www.github.com/wephy/physics-informed-neural-networks

Physics-informed neural networks (PINNs) often require high-accuracy quasi-Newton refinement to obtain reliable partial differential equation solutions, but their residual objectives can exhibit indefinite, nearly singular, and poorly scaled local curvature. Regularized quasi-Newton methods provide established mechanisms for stabilizing secant models, while self-concordant methods provide local-metric rules for curvature-dependent step selection. Building on these two lines of work, we propose SCORE, a self-concordance-inspired quasi-Newton method with decrement-coupled shifted secant geometry for PINN training. Its distinguishing mechanism is that a single quasi-Newton decrement computed from the learned inverse metric jointly determines a strong-Wolfe-tested candidate step and an adaptive shift used to define the next secant geometry. The shifted displacement represents the action of an averaged shifted metric along the accepted step, while requiring neither Hessian construction nor Hessian-vector products. Under a local spectral-equivalence condition, we show that the quasi-Newton decrement and candidate step remain comparable to their counterparts in a positive shifted metric, and recover the normalized self-concordant rule in the matched-metric case. Strong Wolfe acceptance, fallback line search, and standard curvature safeguards provide globalization without modifying the underlying PINN objective. Experiments on the viscous Burgers, Kuramoto--Sivashinsky, Korteweg--de Vries, and complex Ginzburg--Landau equations show that SCORE attains lower final errors than the tested BFGS and self-scaled Broyden baselines. The Burgers ablation further indicates that shifted curvature stabilization and decrement-based step selection make complementary contributions to high-accuracy refinement.

The choice of Riemannian metric can strongly influence the convergence of gradient-based optimization over covariance matrices. Euclidean, Bures-Wasserstein and affine-invariant metrics are common choices, but their relative effectiveness depends on the objective. We introduce a two-parameter family defined by $X^{p}LX^{q}+X^{q}LX^{p}=U$, solved for $L$ at each tangent vector $U$, that contains all three as exact members, at $(0,0)$, $(1,0)$ and $(1,1)$, and extends past them. We treat the choice of member as a particular way of preconditioning for a given problem. To this end, we analyze the conditioning of the Riemannian Hessian at the solution. We show that it obeys a lower bound that depends on $(p,q)$ only through the exponent $r=p+q$. When the Euclidean Hessian is a pure power that mixes no eigendirections, the member $p=q=r/2$ attains that bound, and a closed-form criterion identifies the other members that do. We discuss ways to tune $r$ for a given problem. Experiments on real covariance data confirm the predicted conditioning and the benefit of tuning $r$. A task covariance example shows a further gain from tuning the shape.