Physics-Informed Neural Networks: Blending Equations and Deep Learning for Next-Gen AI

Latest 41 papers on physics-informed neural networks: Aug. 17, 2025

Physics-Informed Neural Networks (PINNs) are rapidly becoming a cornerstone in scientific machine learning, bridging the gap between data-driven AI and fundamental physical laws. By embedding governing equations directly into the neural network’s loss function, PINNs offer a powerful approach to solve complex scientific and engineering problems with unparalleled accuracy, interpretability, and generalization. This isn’t just a niche application; it’s a paradigm shift, enabling robust modeling in areas where traditional numerical methods struggle with high dimensionality, sparse data, or complex geometries. Recent breakthroughs highlight how PINNs are evolving, from enhancing training stability and accuracy to tackling real-world challenges in fields as diverse as fluid dynamics, healthcare, and advanced manufacturing.

The Big Idea(s) & Core Innovations

The latest research in PINNs is tackling fundamental challenges to unlock even greater potential. A recurring theme is the improvement of training stability and accuracy. For instance, the paper “Enhancing Stability of Physics-Informed Neural Network Training Through Saddle-Point Reformulation” introduces a novel saddle-point formulation, addressing the imbalance in loss contributions that often plague PINN training. Similarly, “A matrix preconditioning framework for physics-informed neural networks based on adjoint method” by Song, Wang, Jagtap, and Karniadakis significantly improves convergence and stability by tackling ill-conditioning through matrix preconditioning, enabling robust solutions for challenging multi-scale PDEs like Navier–Stokes.

Addressing data scarcity and extrapolation is another key focus. The “Quantifying data needs in surrogate modeling for flow fields in 2D stirred tanks with physics-informed neural networks (PINNs)” study demonstrates that PINNs can achieve high accuracy with minimal labeled data, making them ideal for expensive data collection scenarios. “Improving physics-informed neural network extrapolation via transfer learning and adaptive activation functions” by Papastathopoulos-Katsaros, Stavrianidi, and Liu further enhances PINN extrapolation capabilities with minimal training cost, reducing errors significantly.

Several papers push the boundaries of accuracy and efficiency. “Breaking the Precision Ceiling in Physics-Informed Neural Networks: A Hybrid Fourier-Neural Architecture for Ultra-High Accuracy” achieves an unprecedented L2 error of 1.94×10−7 for the Euler-Bernoulli beam equation by combining Fourier series with deep neural networks. In a similar vein, “Separated-Variable Spectral Neural Networks: A Physics-Informed Learning Approach for High-Frequency PDEs” introduces SV-SNN, which mitigates spectral bias for high-frequency PDEs, achieving 1-3 orders of magnitude improvement in accuracy. For complex geometries, “Solved in Unit Domain: JacobiNet for Differentiable Coordinate Transformations” introduces JacobiNet, a network that learns continuous, differentiable mappings, dramatically improving accuracy in irregular physical domains.

Beyond numerical precision, PINNs are finding their way into complex real-world applications. The “Generalising Traffic Forecasting to Regions without Traffic Observations” paper introduces GenCast, a model that leverages physics (LWR equation) and external signals (weather) to forecast traffic in data-sparse regions. In healthcare, “Exploration of Hepatitis B Virus Infection Dynamics through Physics-Informed Deep Learning Approach” and “Estimation of Hemodynamic Parameters via Physics Informed Neural Networks including Hematocrit Dependent Rheology” show how Disease Informed Neural Networks (DINNs) and PINNs can model HBV infection and estimate hemodynamic parameters from MRI data, respectively, even with sparse or noisy inputs. Industrially, “Improved Training Strategies for Physics-Informed Neural Networks using Real Experimental Data in Aluminum Spot Welding” integrates real experimental data with PINNs for aluminum spot welding, enhancing accuracy and predictive power in complex manufacturing processes.

Under the Hood: Models, Datasets, & Benchmarks

The advancements in PINNs are underpinned by innovative architectural designs, training strategies, and problem-specific adaptations:

• Architectural Enhancements:
  • LNN–PINN from Ze Tao, Hanxuan Wang, and Fujun Liu introduces liquid residual blocks, boosting predictive accuracy without supervised data (LNN-PINN: A Unified Physics-Only Training Framework with Liquid Residual Blocks).
  • BubbleONet by Yunhao Zhang, Lin Cheng, et al. utilizes adaptive activation functions (Rowdy) within a PI-DeepONet framework to handle high-frequency bubble dynamics (BubbleONet: A Physics-Informed Neural Operator for High-Frequency Bubble Dynamics). Code available at https://github.com/DeepONet/DeepONet and https://github.com/physics-informed-machine-learning/PINN.
  • BridgeNet by Elmira Mirzabeigi, Rezvan Salehi, and Kourosh Parand combines CNNs with PINNs for high-dimensional Fokker–Planck equations, achieving superior accuracy and stability (BridgeNet: A Hybrid, Physics-Informed Machine Learning Framework for Solving High-Dimensional Fokker-Planck Equations).
  • QCPINN from Afrah Farea, Saiful Khan, and Mustafa Serdar Celebi introduces a quantum-classical hybrid for PDEs, significantly reducing parameters while maintaining accuracy (QCPINN: Quantum-Classical Physics-Informed Neural Networks for Solving PDEs). Code at https://github.com/afrah/QCPINN.
  • PIHKAN (Physics-Informed Holomorphic Kolmogorov-Arnold Network) by Matteo Calaf`a, Tito Andriolli, et al. leverages holomorphic neural networks for elliptic PDEs on complex 2D domains (A holomorphic Kolmogorov-Arnold network framework for solving elliptic problems on arbitrary 2D domains).
  • GON by Jianghang Gu, Ling Wen, et al. uses a binary-structured neural network to approximate Green’s functions for interpretable PDE solutions in 3D domains (An explainable operator approximation framework under the guideline of Green’s function). Code at https://github.com/hangjianggu/GreensONet.
• Optimization & Training Strategies:
  • SSBE-PINN by Chen and Xiang introduces a Sobolev Boundary Scheme for H1 convergence in elliptic/parabolic PDEs, ensuring robust and accurate derivatives (SSBE-PINN: A Sobolev Boundary Scheme Boosting Stability and Accuracy in Elliptic/Parabolic PDE Learning). Code available at https://github.com/CChenck/H1Boundary.
  • Adaptive Collocation Point Strategies using QR-DEIM, by Adrian Celaya, David Fuentes, and Beatrice Riviere, improve accuracy by adaptively sampling points in high-gradient regions (Adaptive Collocation Point Strategies For Physics Informed Neural Networks via the QR Discrete Empirical Interpolation Method).
  • DLRS (Dynamic Learning Rate Scheduler) by Veerababu Dharanalakota, Ashwin Arvind Raikar, and Prasanta Kumar Ghosh adapts learning rates based on loss values, enhancing convergence for PINNs and image classification (Improving Neural Network Training using Dynamic Learning Rate Schedule for PINNs and Image Classification). Code at https://github.com/Veerababu-Dharanalakota/DLRS and https://github.com/Ashwin-Aravind-Raikar/DLRS.
  • A Residual Guided strategy with Generative Adversarial Networks for Physics-Informed Transformers by Ziyang Zhang et al. uses GANs, causal masking, and adaptive sampling for state-of-the-art accuracy in nonlinear PDEs (A Residual Guided strategy with Generative Adversarial Networks in training Physics-Informed Transformer Networks). Code: https://github.com/macroni0321/PhyTF-GAN.
  • SiGMoID by Hyunwoo Choa, Hyeontae Job, and Hyung Ju Hwang integrates HyperPINN and Wasserstein GANs for robust inference in dynamic systems from imperfect data (Learning from Imperfect Data: Robust Inference of Dynamic Systems using Simulation-based Generative Model). Code at https://github.com/CHWmath/SiGMoID.
  • Convolution-weighting method for PINNs by Chenhao Si and Ming Yan uses convolution-based weighting and residual-driven resampling for better loss balancing and accuracy (Convolution-weighting method for the physics-informed neural network: A Primal-Dual Optimization Perspective). Code: https://github.com/Shengfeng233/PINN-for-NS-equation.
  • The “Overcoming the Loss Conditioning Bottleneck in Optimization-Based PDE Solvers: A Novel Well-Conditioned Loss Function” paper proposes the Stabilized Gradient Residual (SGR) loss, which directly uses PDE residuals as gradients, significantly accelerating convergence. Code: https://github.com/Cao/WenBo/StabilizedGradientResidual.
• Uncertainty Quantification & Geometry:
  • LVM-GP by Xiaodong Feng et al. provides uncertainty-aware PDE solving by merging latent variable models with Gaussian processes (LVM-GP: Uncertainty-Aware PDE Solver via coupling latent variable model and Gaussian process).
  • GeoHNNs (Geometric Hamiltonian Neural Networks) by Amine Mohamed Aboussalah and Abdessalam Ed-dib explicitly encodes geometric priors for improved stability and energy conservation in physical systems (GeoHNNs: Geometric Hamiltonian Neural Networks).
• Specialized Applications:
  • PVD-ONet by Tiantian Sun and Jian Zu combines DeepONet with Van Dyke matching for multi-scale boundary layer problems, providing fast predictions without retraining (PVD-ONet: A Multi-scale Neural Operator Method for Singularly Perturbed Boundary Layer Problems).
  • For high-dimensional control problems, “Physics-informed approach for exploratory Hamilton–Jacobi–Bellman equations via policy iterations” and “Solving nonconvex Hamilton–Jacobi–Isaacs equations with PINN-based policy iteration” leverage PINN-based policy iteration for scalable and accurate solutions to HJB and HJI equations, respectively.
  • The “Extended Interface Physics-Informed Neural Networks Method for Moving Interface Problems” paper and “Learning Fluid-Structure Interaction Dynamics with Physics-Informed Neural Networks and Immersed Boundary Methods” (code at https://github.com/afrah/pinn_fsi_ibm) tackle dynamic and fluid-structure interaction problems, respectively, demonstrating PINN’s prowess in complex, evolving systems.

Impact & The Road Ahead

These recent advancements highlight a dramatic surge in the capabilities and applications of Physics-Informed Neural Networks. The core impact lies in their ability to solve complex differential equations with greater accuracy, stability, and efficiency, especially in scenarios with sparse or noisy data. This enables more robust scientific discovery, as seen in “DEM-NeRF: A Neuro-Symbolic Method for Scientific Discovery through Physics-Informed Simulation,” which integrates symbolic reasoning into neural networks for interpretable AI models.

The push for higher precision, as discussed in “Challenges in automatic differentiation and numerical integration in physics-informed neural networks modelling,” underscores the growing maturity of the field and the need for robust numerical practices. Furthermore, theoretical breakthroughs like those in “Optimization and generalization analysis for two-layer physics-informed neural networks without over-parametrization” and “Convergence of Implicit Gradient Descent for Training Two-Layer Physics-Informed Neural Networks” provide the foundational understanding necessary for scalable and reliable PINN deployment.

Looking ahead, PINNs are poised to revolutionize scientific computing and engineering. The ability to generalize to unobserved regions, handle complex geometries, and operate with minimal data points makes them invaluable for fields ranging from climate modeling and materials science to personalized medicine and autonomous systems. As research continues to refine their training strategies, address computational bottlenecks, and explore novel architectures, PINNs will undoubtedly unlock new frontiers in our ability to understand, predict, and control the physical world.