train physics-informed networks

This work addresses the lack of a unified understanding of the design principles, applicability, and performance differences between Physics-Informed Neural Networks (PINNs) and Neural Operators (NOs), which hinders the development of reliable data-driven PDE solvers. It proposes the first unified analytical framework that systematically characterizes the design space of both approaches along three dimensions: learning objectives, mechanisms for embedding physical structure, and strategies for computational load distribution. By elucidating the intrinsic connections and fundamental distinctions between these methods, the study not only clarifies the positioning and performance origins of existing techniques but also provides theoretical guidance and novel pathways for designing efficient and robust PDE solvers that effectively integrate physical priors with data-driven learning.

Neural solvers for partial differential equations (PDEs) suffer from high sensitivity to hyperparameters—e.g., collocation point distribution and loss weighting—while classical numerical methods (e.g., finite element methods) require fine meshes, struggle to integrate sparse measurements and physical priors, and necessitate re-computation upon parameter changes. Method: We propose Scientific Constraint Learning (SCL), a framework that reformulates PDE solving as a worst-case constrained learning problem equivalent to the weak formulation. SCL unifies physical laws, symmetry priors, and sparse observations via variational principles, Lagrange multiplier–based optimization, and weak-form neural parameterization—eliminating explicit collocation points, manual loss weighting, and hyperparameter tuning. Results: Experiments demonstrate that SCL achieves high accuracy and strong generalization across diverse PDEs, with computational cost often lower than traditional solvers.

Physics-informed neural networks (PINNs) face challenges—including ill-conditioned optimization, slow convergence, and poor generalization—when solving parametric partial differential equations (PDEs). This paper proposes a data-driven neural solver that parameterizes adaptive gradient descent as a neural network, jointly modeling distributions of PDE coefficients and initial/boundary conditions under physical constraints, while dynamically conditioning the optimizer to alleviate loss function ill-conditioning. To our knowledge, this is the first work to introduce neural solvers into parametric PDE settings, enabling end-to-end training via implicit differentiation and backpropagation. Experiments demonstrate a 2–5× speedup in training with enhanced convergence stability. At inference, the solver generalizes robustly to unseen parameter combinations, significantly reducing required iterations while maintaining high accuracy.

This work proposes NewPINNs, a novel framework that addresses the well-known optimization failures of traditional physics-informed neural networks (PINNs) when solving partial differential equations (PDEs), which often stem from ill-conditioned residual loss formulations, sensitivity to loss weighting, and challenges posed by stiffness or strong nonlinearity. NewPINNs uniquely integrates classical numerical solvers—such as finite volume, finite element, and spectral methods—directly into the neural network training process. Instead of relying on explicit PDE residual and boundary condition losses, the method enforces consistency between the neural network predictions and the states evolved by the embedded numerical solver through a pull-push interaction mechanism. This approach effectively circumvents common failure modes of PINNs, significantly enhancing solution stability and accuracy across a range of forward and inverse PDE problems, particularly in stiff and highly nonlinear regimes.

Sensitivity analysis for steady-state heat conduction in heterogeneous materials—characterized by strong phase contrast and temperature-dependent properties—is computationally expensive when performed via conventional adjoint methods. Method: This paper proposes the Finite Operator Learning (FOL) framework, which tightly integrates neural operators with finite element discretization. FOL embeds physical constraints—including the weak-form energy functional, boundary conditions, and residual stationarity—into a multi-objective loss function, and combines Sobolev-norm training with feedforward networks to jointly predict both PDE solutions and their sensitivities to design parameters in an end-to-end manner. Contribution/Results: FOL requires neither labeled training data nor adjoint computations, ensuring strong physics consistency. It directly outputs high-fidelity solutions and accurate gradients, enabling tangent-matrix-driven microstructural thermal optimization. By eliminating iterative adjoint solves, FOL significantly reduces sensitivity analysis cost while preserving numerical robustness and physical fidelity.

This work addresses the limited data efficiency and poor out-of-distribution (OOD) generalization of existing neural operator methods, which often neglect underlying physical principles—particularly when facing parameter variations or simulation-to-reality transfer. To overcome these limitations, we propose a multi-physics joint training framework that explicitly integrates the original partial differential equations (PDEs) with their simplified canonical forms directly into the neural operator training process. This architecture-agnostic approach is compatible with diverse neural operator designs and consistently enhances model robustness under parameter shifts and cross-domain scenarios. Extensive experiments across multiple 1D, 2D, and 3D PDE tasks demonstrate significant reductions in normalized root mean square error (nRMSE), confirming improved data efficiency and superior OOD generalization performance.

This work addresses a critical gap in existing physics-informed neural network (PINN) tutorials, which often rely on automatic differentiation libraries that obscure the underlying algebraic and gradient propagation mechanisms. Using a first-order initial value problem as a case study, the authors present—for the first time—a complete hand-derived walkthrough of PINN training for a 1-3-3-1 multilayer perceptron with 22 trainable parameters. The exposition covers forward propagation, construction of the composite loss (combining ODE residual and initial condition), backpropagation, and parameter updates, alongside recursive sensitivity relations generalizable to arbitrarily deep networks. Through a Jupyter/PyTorch implementation, the authors validate each manual computation against automatic differentiation and demonstrate that training solely with physics-informed loss—without any ground-truth solution data—achieves a relative L² error of 4.290×10⁻⁴, thereby elucidating the fundamental optimization dynamics of PINNs.

This work addresses the challenge that existing neural operators struggle to simultaneously maintain physical fidelity, generalization capability, and inference efficiency under varying PDE parameters and boundary conditions. The authors propose a generalized neural operator framework that explicitly embeds PDE parameters and boundary conditions to ensure well-posedness and enable efficient cross-domain solutions. Key innovations include a parameter-gated kernel mixture mechanism, a generalized boundary transfer operator, and a unified Dirichlet latent-space representation. A physics-constrained training objective is designed to eliminate instance-specific optimization. Experiments demonstrate that the proposed method significantly outperforms current neural operators across diverse heterogeneous physical scenarios, achieving high accuracy while matching the inference speed of conventional numerical solvers, thereby offering a balanced combination of universality, efficiency, and physical consistency.

This work addresses the challenge of generalizing partial differential equation (PDE) solvers to unseen geometric domains by proposing Geo-NeW, a novel method that jointly learns differential operators and compatible reduced finite element spaces within the framework of finite element exterior calculus to rigorously preserve physical conservation laws. Geo-NeW introduces geometry-aware neural Whitney forms that embed mesh geometric information into both Transformer encodings and basis function construction, thereby endowing neural PDE solvers with strong structure-preserving inductive biases. Furthermore, it devises a new constitutive model parameterization that guarantees the existence and uniqueness of solutions. Evaluated on multiple steady-state PDE benchmarks, the method achieves state-of-the-art performance and significantly outperforms conventional approaches on out-of-distribution geometries.

Physics-informed neural networks (PINNs) enforce partial differential equations (PDEs) via soft constraints, failing to guarantee conservation of linear and quadratic integral quantities—compromising physical consistency and numerical accuracy. This work proposes a novel projection-based method that strictly enforces either independent or joint conservation of these integrals during training. By formulating and solving a nonlinear constrained optimization problem, we derive an explicit projection operator that orthogonally projects the neural network output onto the corresponding conservation manifold in real time. To our knowledge, this is the first approach enabling configurable, simultaneous control of both linear and quadratic integral conservation. The method significantly improves the condition number of the loss landscape, enhancing training stability and convergence speed. Experiments demonstrate reductions in conservation error by three to four orders of magnitude, accompanied by commensurate decreases in PDE solution error, markedly improving physical fidelity and generalization capability.