This work introduces a variational boosting framework in which solutions are constructed additively in function space and separates global nonlinear refinement into a sequence of well-conditioned subproblems while preserving the full variational structure of the operator.
Abstract
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.
This work proposes the Fourier-enhanced alternating Levenberg--Marquardt PINN (FALM-PINN), an optimization framework that decouples representation learning from coefficient fitting within a single nonconvex optimization objective.
Yulun Wu, Matthieu Barreau, Miguel Aguiar et al.· 0 citations
The Physics-Informed Stochastic Configuration Machine is proposed, a novel backpropagation-free framework for both forward and inverse problems in differential equations that achieves high-fidelity predictive accuracy and robust parameter identification while accelerating the training process by orders of magnitude compared to standard PINNs.
Yueze Song, Zhongzhe Chen, Lihui Cen et al.· 0 citations
Experiments show that FC-VPINN achieves approximately one-order-of-magnitude lower prediction errors than the traditional PINN and reduces memory usage to 40% of that required by the baseline, demonstrating improved accuracy and computational efficiency in multidimensional problems.
Wenjie Zhang, Yubo Li, Weidong Cui et al.· Chinese Physics B· 0 citations
Physics-informed neural networks (PINNs) have emerged as a powerful paradigm for solving partial differential equations (PDEs) by embedding governing physical laws into deep neural networks. However, their reliance on computationally expensive gradient-based optimization and deep architectures often results in slow training, high computational cost, and limited scalability. In this work, we propose a novel physics-informed broad learning system (PI-BLS), the first physics-informed learning framework based on broad RdNNs. The proposed formulation embeds the governing differential operator and the associated initial and boundary constraints directly into a linear output-layer optimization problem, thereby replacing nonlinear gradient-based training with a deterministic least-squares solution obtained via the pseudoinverse. Consequently, the entire learning process is reduced to a single linear optimization stage while preserving the underlying physical constraints. As a result, PI-BLS offers an efficient learning paradigm for a physics-informed learning framework for solving PDEs that eliminates iterative backpropagation while preserving the underlying physical constraints. Experimental results on representative forward PDE benchmarks demonstrate that PI-BLS achieves competitive and often superior performance with reduced training time and model parameters compared with conventional PINNs.
Pinki Khatun, M. Sajid, Abhinav Jha et al.· 0 citations
We introduce a physics-informed framework for learning finite-dimensional embeddings of solution families of partial differential equations. The method uses a multihead Physics-Informed Neural Network in which a shared body learns a latent manifold representing the solution space, while linear heads reconstruct individual solutions associated with different initial conditions. A head-orthogonalization penalty removes degeneracies in the latent representation and stabilizes the principal-component spectrum across training realizations. Because the initial condition is built into the network output by construction, these principal components measure the additional variability the network learns on top of the initial profile, not the full solution itself. We apply the method to the one-dimensional viscous Burgers equation, with the heat and wave equations as robustness checks. For a latent dimension $n_b=20$, the learned manifolds exhibit pronounced effective dimensional reduction: for Burgers dynamics, only $2$-$4$ principal components capture about $95\%$ of the latent-space variance, while $4$-$7$ capture about $99\%$, depending on the initial-condition family; the same qualitative compression holds for the heat and wave equations. We also split the wavenumber axis into bands (``Fourier shells'') and measure how much each band contributes to every principal component. The resulting frequency profile is invariant under the change-of-basis freedom that the orthogonalization penalty leaves in the latent space, and is therefore reproducible across independent training runs. More broadly, this establishes the learned spectral profiles and principal components as robust observables of solution-manifold geometry.
Raúl Jiménez, S. Mayboroda, P. Protopapas et al.· 0 citations
The universal consistency of PIKS is established for linear differential constraints, proving that for universal kernels (such as Gaussian or Mat\'ern), the estimator asymptotically learns the target while satisfying physical constraints.
Joachim Bona-Pellissier, Giacomo Meanti, Matteo Santacesaria et al.· 0 citations