physics-informed learning

In scientific machine learning, learned features often lack physical interpretability and meaningful mechanistic grounding. Method: We propose a physics-informed nonlinear feature construction paradigm that integrates dimensional analysis with domain-specific physical constraints to enforce physical consistency in the feature mapping; it employs feature importance ranking to identify dominant physical mechanisms and supports governing-equation discovery and novel physical relation inference when first-principles laws are unknown. Contribution/Results: Experiments across multiple scientific datasets demonstrate significant improvements in regression accuracy and classification skill scores, while simultaneously achieving enhanced interpretability and mechanistic insight. The framework provides a generalizable, interpretable machine learning approach for scientific discovery—bridging data-driven modeling with physical understanding.

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.

Physics-informed convolutional neural networks (PICNNs) suffer from heavy reliance on manual design, poor generalizability, and limited adaptability across diverse partial differential equation (PDE) problems. Method: We propose the first AutoML framework for PICNNs, featuring a two-stage neural architecture search (NAS) that jointly optimizes CNN architectures and physics-informed loss functions. Specifically, we construct a learnable loss-function factor space and residual adjustment operators, and integrate them with a physics-constrained CNN architecture search space. Contribution/Results: This work pioneers co-automated design of both network topology and physical loss components. Evaluated on multiple PDE benchmarks, our method achieves significantly faster convergence, higher prediction accuracy, and superior cross-problem generalization compared to handcrafted PICNNs—effectively addressing the long-standing challenge of coupled optimization between model architecture and physics-based loss functions in physics-driven learning.

Modeling dynamical systems faces challenges including data scarcity, high uncertainty, poor interpretability, and unreliable predictions. Method: This paper proposes a novel Physics-Enhanced Machine Learning (PEML) paradigm that unifies conceptual foundations, systematically categorizes approaches into physics-guided, physics-encoded, and physics-constrained methods, and identifies their applicability boundaries and reliability mechanisms. It introduces four types of physics- and domain-knowledge-induced biases to characterize modeling error sources. Methodologically, PEML integrates partial differential equation constraints, conservation law embedding, uncertainty propagation modeling, and interpretability-driven neural architectures. Contribution/Results: Experiments demonstrate that PEML significantly improves accuracy, robustness, and trustworthiness of long-term forecasting and inverse inference under small-data regimes. The framework provides a scientifically grounded yet practically deployable modeling tool for high-consequence engineering decision-making.

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.

This work addresses the challenge of efficiently adapting pre-trained foundation models for partial differential equations (PDEs) under data scarcity and distribution shift. We propose a physics-informed fine-tuning framework that incorporates physical constraints—such as PDE residuals and boundary conditions—directly into the fine-tuning objective, enabling data-efficient adaptation without requiring ground-truth solutions. To our knowledge, this is the first systematic demonstration of the effectiveness of physics-informed fine-tuning for transferring PDE foundation models. By integrating a hybrid fine-tuning strategy, our approach significantly enhances out-of-distribution generalization. Experiments show that, even in the absence of ground-truth solutions, our method achieves accuracy comparable to purely data-driven approaches on unseen PDE tasks and consistently outperforms them when only limited data are available.

This work addresses key limitations of conventional physics-informed neural networks (PINNs) in solving differential equations—namely parameter redundancy, weak locality, and insufficient control over solution smoothness. The authors propose Physics-Informed Splines (PI-Splines), which directly parameterize the unknown field using tensor-product B-splines with trainable control points. Retaining the residual-driven training paradigm of PINNs, PI-Splines inherently offer compact support, explicit smoothness control, and analytically computable derivatives. By strongly enforcing boundary conditions, the method endows spline parameters with clear geometric meaning while leveraging structured representation and efficient optimization strategies. Experimental results demonstrate that PI-Splines achieve stable and efficient performance across multiple benchmark problems, significantly outperforming traditional neural network architectures in terms of parameter efficiency, locality, and representational capacity.

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 enforcing physical symmetries—such as rotational equivariance—in machine learning models without imposing explicit architectural constraints. The authors propose a general, architecture-agnostic approach that introduces a novel metric to quantify the degree of symmetry learning, employs spectral analysis to diagnose failure modes, and leverages targeted data augmentation to guide unconstrained Transformers—including graph neural networks and PointNet-style architectures—to progressively approximate equivariance across layers during training. Experiments demonstrate that injecting only the minimal necessary inductive bias substantially enhances physical fidelity, numerical stability, and predictive accuracy, while preserving the model’s expressive capacity.