Physics-Informed Neural Networks: Navigating the Future of Scientific AI with Breakthroughs in Robustness, Efficiency, and Interpretability

Latest 50 papers on physics-informed neural networks: Nov. 2, 2025

Physics-Informed Neural Networks (PINNs) are rapidly transforming scientific computing by integrating physical laws directly into deep learning models. This powerful paradigm allows AI to not only learn from data but also adhere to fundamental scientific principles, addressing complex challenges in fields from engineering to biomedical science. However, PINNs face hurdles like spectral bias, computational cost, and ensuring solution robustness, especially in noisy or high-dimensional scenarios. Recent research is pushing the boundaries, delivering innovative solutions that enhance accuracy, efficiency, and interpretability, making PINNs an even more compelling tool for scientific discovery.

The Big Idea(s) & Core Innovations

The latest advancements in PINNs are centered on making these models more reliable, faster, and applicable to a wider array of real-world problems. A key theme involves refining how physical constraints are integrated and optimized. For instance, the paper Incorporating Local H”older Regularity into PINNs for Solving Elliptic PDEs by Qirui Zhou, Jiebao Sun, Yi Ran, and Boying Wu from the School of Mathematics, Harbin Institute of Technology, introduces local Hölder regularity into PINN loss functions. This provides a principled way to embed smoothness assumptions, improving accuracy and robustness for elliptic PDEs through variable-distance sampling. Complementing this, Enforcing boundary conditions for physics-informed neural operators by N. G¨OSCHEL et al. proposes a robust framework for integrating domain-specific boundary constraints, critical for the accuracy of physics-informed neural operators, particularly in complex systems like Navier-Stokes equations.

Efficiency and scalability are also major drivers. The PIELM framework, explored in A Rapid Physics-Informed Machine Learning Framework Based on Extreme Learning Machine for Inverse Stefan Problems by Professor Pei-Zhi Zhuang et al. from Shandong University, leverages extreme learning machines (ELMs) to dramatically reduce training time (over 94%) and improve accuracy by several orders of magnitude for inverse Stefan problems. Similarly, Towards Fast Option Pricing PDE Solvers Powered by PIELM further demonstrates PIELM’s capability to solve financial PDEs up to 30 times faster than traditional PINNs, making real-time applications feasible. Addressing the scalability of complex optimizations, Improving Energy Natural Gradient Descent through Woodbury, Momentum, and Randomization by Andrés Guzmán-Cordero et al. from the Vector Institute introduces Woodbury’s identity and a momentum scheme to accelerate natural gradient descent in PINNs, drastically cutting computational costs.

Addressing the notorious spectral bias—PINNs’ struggle with high-frequency components—is another critical area. Iterative Training of Physics-Informed Neural Networks with Fourier-enhanced Features by Yulun Wu et al. from KTH Royal Institute of Technology, presents IFeF-PINN, which mitigates this bias using Random Fourier Features and an iterative two-stage training algorithm, significantly improving high-frequency PDE approximation. In parallel, Physics-Informed Neural Networks with Fourier Features and Attention-Driven Decoding by Rohan Arni and Carlos Blanco (High Technology High School, Lincroft, NJ and The Pennsylvania State University) introduces S-Pformer, an encoder-free Transformer-based PINN that combines Fourier feature embeddings and attention mechanisms for better efficiency and accuracy.

For overcoming the challenges of complex geometries and dynamic systems, Meshless solutions of PDE inverse problems on irregular geometries by Author A and Author B (University of Example and Institute of Advanced Research) introduces meshless methods for PDE inverse problems on non-smooth domains, enhancing accuracy and efficiency. For time-dependent problems, Frozen-PINN: Fast training of accurate physics-informed neural networks without gradient descent by Chinmay Datar et al. from the Technical University of Munich presents a revolutionary gradient-descent-free approach, achieving up to 100,000x faster training times by using space-time separation and random features.

Several papers also focus on enhancing the robustness and interpretability of PINNs. Uncertainty-Aware Diagnostics for Physics-Informed Machine Learning by Mara Daniels et al. (Massachusetts Institute of Technology, University of Melbourne, and University of California at Berkeley) proposes the Physics-Informed Log Evidence (PILE) score, a novel metric for uncertainty-aware model selection even without data. For robust control, Lyapunov-Based Physics-Informed Deep Neural Networks with Skew Symmetry Considerations by Rebecca G. Hart et al. (University of Florida and Air Force Research Laboratory) develops a PINN controller for Euler-Lagrange systems that incorporates skew-symmetry properties, improving approximation capabilities by nearly 20%. The AW-EL-PINNs: A Multi-Task Learning Physics-Informed Neural Network for Euler-Lagrange Systems in Optimal Control Problems by Chuandong Li and Runtian Zeng from Southwest University proposes an adaptive loss weighting mechanism for multi-task PINNs, enhancing stability and accuracy in optimal control problems.

Finally, for integrating PINNs into broader scientific and engineering workflows, Lang-PINN: From Language to Physics-Informed Neural Networks via a Multi-Agent Framework by Xin He et al. from A*STAR, Hong Kong Baptist University, and North China Electric Power University, introduces an LLM-driven multi-agent system that automates the generation of PINNs from natural language, significantly reducing manual effort. StruSR: Structure-Aware Symbolic Regression with Physics-Informed Taylor Guidance by Yunpeng Gong et al. (Xiamen University, Université de Sherbrooke) integrates physics-informed priors from PINNs into symbolic regression, leveraging Taylor expansions to discover interpretable mathematical expressions with structural and physical fidelity.

Under the Hood: Models, Datasets, & Benchmarks

Recent work has introduced or heavily utilized several key models and strategies:

• PINN Architectures & Enhancements:
  • IFeF-PINN: An iterative two-stage training algorithm that uses Random Fourier Features to mitigate spectral bias in PINNs. (https://arxiv.org/pdf/2510.19399)
  • S-Pformer: A transformer-based PINN architecture with Fourier feature embeddings and attention-driven decoding, significantly reducing parameter count. (https://arxiv.org/pdf/2510.05385)
  • PIELM (Physics-Informed Extreme Learning Machines): Offers significantly faster training and comparable accuracy to traditional PINNs for various PDEs, including inverse Stefan problems and financial option pricing. (https://arxiv.org/pdf/2510.21426, https://arxiv.org/pdf/2510.04322)
  • Frozen-PINN: A gradient-descent-free PINN training approach for time-dependent PDEs using space-time separation and random features for 100,000x speedups. Code available at https://gitlab.com/felix.dietrich/swimpde-paper.git. (https://arxiv.org/pdf/2405.20836)
  • LieSolver: A PDE-constrained solver that enforces exact solutions of IBVPs using Lie symmetries, outperforming PINNs in convergence and interpretability. Code available at https://github.com/oduwancheekee/liesolver. (https://arxiv.org/pdf/2510.25731)
  • AB-PINNs (Adaptive-Basis PINNs): Uses residual-driven domain decomposition with adaptive basis functions and a global network for multiscale PDEs. (https://arxiv.org/pdf/2510.08924)
  • PO-CKAN (Physics Informed Deep Operator Kolmogorov Arnold Networks): Integrates rational KAN modules into DeepONets for efficient PDE solving, achieving superior accuracy-efficiency tradeoffs. (https://arxiv.org/pdf/2510.08795)
  • AW-EL-PINNs: A multi-task learning PINN framework for Euler-Lagrange systems in optimal control problems, using adaptive loss weighting. (https://arxiv.org/pdf/2509.25262)
  • dfNNs (Divergence-Free Neural Networks): Enforces exact mass conservation in ice flow modeling, outperforming PINNs in climate applications. Code available at https://github.com/kimbente/mass_conservation_on_rails. (https://arxiv.org/pdf/2510.06286)
  • THINNs (Thermodynamically Informed Neural Networks): Replaces ad-hoc L2 penalization with a rate functional derived from large deviation principles for more physically consistent PDE solutions. (https://arxiv.org/pdf/2509.19467)
• Optimization & Training Strategies:
  • AutoBalance: A post-combine framework that uses independent optimizers per loss component to effectively balance gradients in PINN training. (https://arxiv.org/pdf/2510.06684)
  • AMStraMGRAM: An adaptive multi-cutoff strategy that enhances natural gradient methods for PINNs, leveraging a ‘flattening phenomenon’ for improved L2 error. Code available at https://anonymous.4open.science/r/AMStraMGRAM-8D1B/. (https://arxiv.org/pdf/2510.15998)
  • Nyström-Accelerated Primal LS-SVMs: Reduces computational complexity for ODEs, achieving up to 6000x speedups over PINNs and LS-SVMs. Code available at https://github.com/AI4SciCompLab/NLS-SVMs. (https://arxiv.org/pdf/2510.04094)
  • GLF (Global–Local Fusion) Sampling: A residual-adaptive neighborhood sampling method for PINNs, improving both accuracy and efficiency. (https://arxiv.org/pdf/2510.24026)
  • PINN BALLS: Scales second-order optimization methods for PINNs using domain decomposition and Adversarial Adaptive Sampling (AAS). (https://arxiv.org/pdf/2510.21262)
• Benchmarking & Evaluation:
  • Morpheus: A benchmark for video generative models, evaluating physical reasoning using real physical experiments and explicit conservation laws. Code and resources at https://physics-from-video.github.io/morpheus-bench/. (https://arxiv.org/pdf/2504.02918)
  • Comparative PDE Solver Study: Compares PINNs, Deep Ritz Method (DRM), and Weak Adversarial Networks (WANs) on Poisson and Schr”odinger equations. Code available at https://github.com/JiakangC/Neural-Network-Based-PDE-Solver.git. (https://arxiv.org/pdf/2510.09693)
  • COVID-19 in Germany Dataset: Used to model spatio-temporal dynamics with PINNs, with code for data at https://github.com/robert-koch-institut/SARS-CoV-2-Infektionen_in_Deutschland. (https://arxiv.org/pdf/2510.06776)
  • RBF-PIELM Benchmark: Compares RBF-PIELM against traditional PINNs for higher-order PDEs like the biharmonic equation, highlighting faster training. (https://arxiv.org/pdf/2510.04490)
• Integration with traditional methods:
  • Finite Element Method (FEM) Enrichment: Enriching continuous Lagrange finite element approximation spaces using neural networks integrates PINNs into FEM for enhanced PDE solution accuracy, allowing coarser meshes. (https://arxiv.org/pdf/2502.04947)
  • Differentiable Physics: Differentiable physics for sound field reconstruction integrates differentiable PDE solvers into neural network training for sound field reconstruction, outperforming PINNs. Code at https://github.com/samuel-verburg/differentiable-soundfield-reconstruction. (https://arxiv.org/pdf/2510.04459)

Impact & The Road Ahead

These advancements highlight a pivotal moment for physics-informed neural networks. The innovations in regularization, such as local Hölder regularity and robust boundary condition enforcement, are making PINNs more mathematically sound and reliable. The dramatic speed-ups achieved through PIELMs, Frozen-PINN, and optimized gradient descent methods are transforming PINNs from a promising theoretical concept into a practical tool for real-time applications in finance, engineering, and beyond. This enhanced efficiency democratizes access to complex simulations, potentially accelerating discovery in fields previously limited by computational bottlenecks.

The focus on mitigating spectral bias and adapting to complex geometries signifies PINNs’ maturity for tackling intricate scientific problems, from predicting ocean temperatures with fine-tuned atmospheric models (Leveraging an Atmospheric Foundational Model for Subregional Sea Surface Temperature Forecasting) to modeling adoptive cell therapy in cancer (Modeling Adoptive Cell Therapy in Bladder Cancer from Sparse Biological Data using PINNs). The introduction of explainable architectures like PIKAN (PIKAN: Physics-Inspired Kolmogorov-Arnold Networks for Explainable UAV Channel Modelling) and Lang-PINN’s natural language to PINN generation capability point towards a future where scientific AI is not only powerful but also intuitive and accessible to a broader range of researchers and practitioners.

The emphasis on uncertainty quantification, as seen with the PILE score and extended fiducial inference (Uncertainty Quantification for Physics-Informed Neural Networks with Extended Fiducial Inference), is critical for building trust in AI-driven scientific predictions. This robust understanding of model confidence will be vital for sensitive applications like climate modeling and biomedical diagnostics. Furthermore, the integration of evolutionary optimization (Evolutionary Optimization of Physics-Informed Neural Networks: Evo-PINN Frontiers and Opportunities) and multi-agent frameworks signals a move towards autonomous and self-optimizing PINN systems. This heralds a future where AI not only solves complex physical equations but also designs and refines its own solutions, pushing the frontiers of scientific AI and engineering innovation even further.