physics-informed architecture

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.

This work investigates whether intrinsic symmetries in training data induce conserved quantities during gradient flow training of neural networks. By integrating tools from differential geometry and dynamical systems theory, the study establishes—for the first time—a systematic connection between data symmetries and conservation laws in training dynamics, employing tensorized networks (including linear, polynomial, and Lightning Attention architectures) as an analytical framework. The analysis demonstrates that, under general non-polynomial losses, data symmetries do not yield additional conserved quantities; however, when combined with data augmentation under mean squared error (MSE) loss, novel conserved quantities emerge. This finding uncovers a distinctive conservation mechanism specific to MSE loss and offers a new perspective for understanding the dynamics of neural network training.

Deep neural networks (DNNs) suffer from high memory overhead, computational cost, and poor interpretability. Method: This paper proposes Physics Equation Neuralization (PE-Net), a framework that directly constructs differentiable, trainable multilayer architectures from fundamental physical equations—such as the nonlinear Schrödinger equation—replacing black-box DNNs with physics-informed models for data representation learning. Contribution/Results: PE-Net innovatively treats physically meaningful equation parameters as learnable variables for the first time and enables quantitative attribution of term-level physical importance. It integrates differentiable physics simulation, embedded physical constraints, and standard backpropagation. Experiments demonstrate order-of-magnitude parameter reduction (1–2×) in time-series modeling while ensuring full model interpretability. Generalization to the Gross–Pitaevskii equation confirms framework universality, and classification performance is determined by the contribution degree of dominant physical terms.

This study addresses critical challenges in applying deep learning to scientific computing—namely, poor interpretability, heavy data dependency, and insufficient physical consistency—within physics-based simulation scenarios. We propose a physics-driven AI modeling framework integrating physics-informed loss functions, differentiable simulators, diffusion-based generative models, physics-guided reinforcement learning, and custom neural architectures, implemented via an interactive Jupyter-based experimental platform. Crucially, we pioneer the systematic embedding of physical priors across the entire deep learning pipeline—model formulation, training, and inference—enabling high-fidelity, data-efficient, and verifiable scientific modeling. The resulting methodology is modular, reusable, and immediately deployable, significantly enhancing model generalizability and interpretability. This work establishes a novel paradigm and technical foundation for next-generation scientific foundation models.

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 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 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.

Deep learning models exhibit generalization capabilities in real-world tasks that surpass predictions from classical statistical learning theory, yet the underlying mechanisms remain poorly understood. This work addresses this gap by integrating statistical learning theory with physics-inspired priors to systematically analyze neural scaling laws under physically constrained scenarios. It uncovers novel scaling behaviors and explicitly models the interplay between inductive biases and architectural design choices. Leveraging a physics-informed machine learning framework, the study elucidates the statistical mechanisms responsible for the exceptional generalization of deep learning in physics-related tasks, thereby providing theoretical foundations and practical guidance for designing models tailored to scientific computing applications.

Existing neural operators often fail to strictly enforce physical conservation laws—such as mass and energy conservation—leading to degraded accuracy, limited generalization, and reliance on problem-specific architectures in time-dependent PDE solving. To address this, we propose a universal plug-and-play neural operator framework that explicitly models and rigorously enforces conservation constraints via an extensible encoder–decoder module for conserved quantities, operating atop arbitrary backbone neural operators without modifying their internal architecture. This is the first approach to decouple conservation enforcement from data-driven learning, providing theoretical guarantees on conservation preservation while enhancing predictive performance. Experiments across diverse PDEs—including adiabatic systems, the shallow water equations, and the Allen–Cahn equation—demonstrate significant improvements in prediction accuracy, strict adherence to multiple conservation laws, and strong cross-domain generalization capability.

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.