ODE Solving

The Lane-Emden Equation is a class of nonlinear differential equations commonly used to model phenomena in astrophysics and mathematical physics, such as stellar structure and thermal behavior in polytropic gas spheres. This study presents a novel hybrid framework that integrates Physics-Informed Neural Networks (PINNs) with boosting methods to enhance the accuracy and robustness of solutions for Lane-Emden type differential equations. Standard PINNs leverage physical laws encoded within the loss function, but their predictive performance can be constrained by optimization challenges and approximation errors. To address this, we introduce a post-processing boosting stage where the residual errors from the PINN prediction are modeled using some regression algorithms - including LightGBM, Random Forest, XGBoost, and Support Vector Regression (SVR). The outputs from these boosted models are aggregated to generate a corrected, high-fidelity prediction. The proposed methodology is systematically validated through quantitative metrics and comparative analysis against the standard PINN baseline. The results demonstrate that the boosted PINN framework significantly reduces prediction error, and improves generalization, offering a powerful and flexible approach for solving scientific and engineering problems governed by differential equations.

Physics-informed machine learning has gained significant attention in recent years. In regimes of limited data, parametrized Gaussian processes have become popular. Existing approaches, however, often face limitations, such as requiring parametrizable (also called controllable) systems or a large number of output tasks. In this work, we introduce a systematic procedure we call "ghost tasking", using auxiliary tasks to circumvent these limitations. We prove that such ghost tasks can render any non-parametrizable system effectively parametrizable, enabling algorithmic construction of parametrized Gaussian Processes while keeping the number of required tasks (i.e. output dimensions) and latent functions low. We find that ghost tasking performs especially well in an inverse problem setting, even with very few available data. We show the usage and power of ghost tasking in three experiments, providing systematic comparisons to the only other currently available method applicable to all experiments. We provide necessary syntax and explications for two computer algebra programs that compute parametrizations for systems with polynomial or rational coefficients. Our theoretical results extend to systems with meromorphic functions.

Ordinary differential equations (ODEs) underlie models in science and engineering, and many applications need derivatives of their solutions with respect to parameters. Ensembles of independent trajectories suit graphics processing units (GPUs), but current GPU software forces a trade-off: the fastest ensemble solvers cannot be differentiated in reverse mode at the speed they solve, and the solvers built for differentiation solve more slowly. No single tool has yet offered a reverse-mode gradient at the speed of a fused-kernel solve. We present GRADSOLVE, an open-source JAX library for solving and reverse-mode differentiating low-dimensional ODE ensembles on NVIDIA GPUs. It records the steps an adaptive solver accepts and differentiates a fixed-step replay of them; the returned gradient is the exact discrete adjoint of those steps, the same derivative Diffrax returns by default, obtained more cheaply from a fixed-length chain than from an adaptive loop. It targets ensembles differentiated many times against one recorded mesh, keeps Diffrax as a fallback, and supports explicit and Rosenbrock integrators. Used as a solver, GRADSOLVE's forward-only kernel ran 2.8x faster than DiffEqGPU.jl; used for gradients, once a record exists, it computed them 5.6-14.1x faster than Diffrax's checkpointed adjoint at matched forward-state accuracy across three GPU generations, the advantage narrowing on large ensembles and, on stiff systems, down to parity at tight accuracy. GRADSOLVE is released at https://github.com/ECLIPSE-AI4Science/gradsolve.

Recovering explicit symbolic solutions of partial differential equations (PDEs) from physics-informed neural networks (PINNs) is attractive, yet selecting and fitting expressions against neural predictions can lock the recovered solution into the teacher's approximation error. We propose DeSyR (Decoupled Symbolic Recovery) to break this dependence. DeSyR uses PINNs only to discover candidate structures and initial coefficient estimates; once a structure is fixed, the governing equations and constraints alone determine the refined coefficients. Multiple candidates with different fitting-error--complexity trade-offs are retained, evaluated after physics-only refinement, and reduced to the simplest expression within a physically near-optimal set. Joint refinement extends the same principle to coupled fields. For fixed candidate parameterizations, affine error identities and local nonlinear analysis show how teacher errors enter fitted coefficients and establish conditions under which physical residuals control coefficient and solution errors. Across seven benchmarks covering scalar evolution equations, multidimensional elliptic problems and coupled incompressible flows, DeSyR attains mean relative solution errors on the order of $10^{-14}$ or smaller and lower reported mean physical residuals than PINNs, physics-informed symbolic networks and teacher-fitted symbolic distillation. With the expression structure held fixed, estimating coefficients from physical constraints further yields more accurate frequency estimates and solutions than fitting teacher predictions. DeSyR therefore turns neural approximations into guides for structure discovery, while physical constraints determine the final explicit solution.

While Physics-Informed Neural Networks (PINNs) have emerged as a transformative paradigm for solving complex differential equations, their reliance on backpropagation-based gradient descent and automatic differentiation (AD) imposes significant computational bottlenecks and severe non-convex optimization challenges. To overcome these fundamental limitations, we propose the Physics-Informed Stochastic Configuration Machine (PI-SCM), a novel backpropagation-free framework for both forward and inverse problems in differential equations. The core mathematical contribution lies in the analytical evaluation of local Jacobians for nonlinear differential operators, which facilitates a linearized representation of the physical loss and projects it into a unified, linearized algebraic subspace. This reformulation allows for the explicit determination of optimal network weights via a sequence of generalized linear least squares solvers, effectively bypassing the iterative traps of traditional nonlinear optimizers. We develop a progressive algorithmic suite comprising localized construction (PI-SC-I), sliding-window updating (PI-SC-II), and global updating (PI-SC-III), and rigorously establish their universal approximation properties. Extensive experiments demonstrate that PI-SCM achieves high-fidelity predictive accuracy and robust parameter identification while accelerating the training process by orders of magnitude compared to standard PINNs. Our work provides a highly efficient and scalable foundation for next-generation, real-time Scientific Machine Learning applications.

In recent years, driven by the computational demands of data-intensive applications such as artificial intelligence and scientific computing, analog computing has gained renewed interest. Given the diversity of computational tasks and recent advancements in analog CMOS circuits and resistive memory technologies, we refer to the evolving landscape as modern analog computing. In this context, we identify three core computational primitives: solving differential equations, solving matrix equations, and performing matrix-vector multiplications, and we explore the connections among them. We also examine various hardware implementations of these analog computing operators, including those built with discrete components, integrated circuits, and resistive memory devices. Among these, resistive memory arrays emerge as particularly promising due to their implementation efficiency. The paper then surveys recent progress in leveraging modern analog computing to solve differential and matrix equations using both advanced analog CMOS circuits and resistive memory arrays. Finally, we discuss the applications of these circuits, the precision and scalability issues and their potential solutions, the relationship with in-memory computing, and the unique computational complexity of analog computing. This paper provides a unified perspective on analog computing, highlighting its strengths, current developments, and challenges, and positioning it as a pivotal enabler of next-generation computational frontiers.

Filtering-based probabilistic numerical solvers for ordinary differential equations (ODEs) have been established as a flexible and efficient simulation framework with built-in numerical uncertainty quantification. However, problems that are both stiff and high-dimensional remain a challenge, as current methods are either stable and have cubic cost in the ODE dimension, or scale linearly at the expense of stability. In this paper, we close this gap and develop probabilistic ODE solvers that are both stable and scalable. We propose two complementary strategies. First, we develop a matrix-free update step that uses Jacobian-vector products, iterative linear solvers, and stochastic covariance estimation to enable linear scaling, all while retaining stability. Second, we propose iterative re-linearization to further improve stability without sacrificing scalability, turning probabilistic ODE solvers into fully implicit methods. We evaluate the proposed approaches on a range of stiff and high-dimensional problems and demonstrate improved stability and scalability over established probabilistic solvers.

We develop a branched signature kernel solver for linear and nonlinear ordinary differential equations driven by a single observed trajectory of a possibly rough forcing signal--a setting common within earthquake engineering, finance, biology, and structural health monitoring, where only one forcing realization is available, and the solver must respect the underlying physical law without an ensemble of realizations. We first introduce a count-sampling construction method to turn the single observation into a hierarchical family of $N+1$ nested training paths on which the branched signature kernel can be evaluated; this allows the signature kernel machinery, originally designed for multi-realization regression problems, to operate on a single-trajectory observation. Then we build a kernel-collocation framework, which places the ansatz either on the highest-order derivative of the solution or on the solution itself. We prove a universal approximation theorem for the branched signature kernel, leveraging the Hairer--Kelly morphism to express branched signature evaluations through geometric signatures of time-extended paths. The offline solver is extended to a streaming Test/Train/Retrain protocol with optional closed-form online updates in both linear and nonlinear cases. Numerical experiments on six benchmarks show accurate, stable predictions across all regimes.