Solving stiff dark matter equations via Jacobian Normalization with Physics-Informed Neural Networks

Physics-Informed Neural Networks (PINNs) solve differential equations by minimizing the residual of a nonlinear operator over a neural parameterization of the solution. However, monolithic PINNs often suffer from ill-conditioning, spectral bias, and optimization instability. We introduce a variational boosting framework in which solutions are constructed additively in function space. Each stage trains a weak learner whose converged correction satisfies a local orthogonality condition, equivalent to a projected functional gradient descent step onto the tangent space of the network's function manifold. Because each correction network is deliberately small, the restricted minimization admits full Newton or conjugate gradient updates, which are typically infeasible in large PINNs. The resulting method separates global nonlinear refinement into a sequence of well-conditioned subproblems while preserving the full variational structure of the operator. This framework provides a geometric interpretation of multi-stage PINNs as projected functional gradient descent and enables stable second-order optimization for nonlinear differential equations.

Efficient and robust optimization is essential for neural networks, enabling scientific machine learning models to converge rapidly to very high accuracy -- faithfully capturing complex physical behavior governed by differential equations. In this work, we present advanced optimization strategies to accelerate the convergence of physics-informed neural networks (PINNs) for challenging partial (PDEs) and ordinary differential equations (ODEs). Specifically, we provide efficient implementations of the Natural Gradient (NG) optimizer, Self-Scaling BFGS and Broyden optimizers, and demonstrate their performance on problems including the Helmholtz equation, Stokes flow, inviscid Burgers equation, Euler equations for high-speed flows, and stiff ODEs arising in pharmacokinetics and pharmacodynamics. Beyond optimizer development, we also propose new PINN-based methods for solving the inviscid Burgers and Euler equations, and compare the resulting solutions against high-order numerical methods to provide a rigorous and fair assessment. Finally, we address the challenge of scaling these quasi-Newton optimizers for batched training, enabling efficient and scalable solutions for large data-driven problems.

Physics-Informed Neural Networks (PINNs) often struggle to train reliably on stiff and oscillatory dynamical systems due to poor optimization conditioning. While prior work has emphasized representational remedies such as spectral parameterizations, the optimization implications of initial-condition (IC) embeddings in adaptive spectral PINNs have not been well characterized. In this work, we show that the choice of IC gating function induces explicit time-dependent gradient scaling, which interacts with spectral representations during training. Using a nonlinear stiff spring-pendulum ODE as a controlled benchmark, we compare exponential and linear IC gates in combination with fixed and adaptive Fourier spectral trunks. We observe stiffness-dependent changes in relative dominance for adaptive PINNs: at moderate stiffness ($k=20$), exponential gating often yields lower error but exhibits heterogeneous behavior across random seeds, whereas at higher stiffness ($k=60$), linear gating becomes preferable, with additional reversals observed at larger $k$. These trends hold for both relative $L^2$ error and maximum pointwise error and are confirmed by paired Wilcoxon signed-rank tests with Holm correction. Overall, our results demonstrate that IC embeddings are not a neutral design choice in PINNs: the induced gradient scaling materially shapes optimization conditioning in stiff regimes, with distinct sensitivity patterns in baseline and adaptive spectral models.