Physics-Informed Neural Networks: Navigating the Future of Scientific Machine Learning

Latest 50 papers on physics-informed neural networks: Sep. 14, 2025

Physics-InInformed Neural Networks (PINNs) have rapidly emerged as a powerful paradigm, blending the expressive power of deep learning with the foundational principles of physics. This fusion promises to revolutionize how we model, simulate, and understand complex physical systems, from predicting asteroid gravity fields to optimizing cancer treatments. Recent breakthroughs, as highlighted in a flurry of innovative research, are pushing the boundaries of PINN capabilities, addressing long-standing challenges in accuracy, stability, and generalization.

The Big Idea(s) & Core Innovations

At the heart of these advancements lies a concerted effort to overcome limitations like spectral bias, handling discontinuities, and improving computational efficiency. A standout innovation is ReBaNO, proposed by Haolan Zheng, Yanlai Chen, Jiequn Han, and Yue Yu from Massachusetts Dartmouth. In their paper, “ReBaNO: Reduced Basis Neural Operator Mitigating Generalization Gaps and Achieving Discretization Invariance”, they introduce a reduced basis-driven physics-informed neural operator that uniquely achieves strict discretization invariance, significantly reducing generalization gaps for PDE solutions. This is crucial for robust scientific simulations.

Several papers tackle the stability and accuracy challenges head-on. “Neuro-Spectral Architectures for Causal Physics-Informed Networks” by Arthur Bizzi and colleagues from EPFL, IMPA, and UERJ presents NeuSA, which integrates spectral methods to overcome spectral bias and causality issues, leading to faster convergence and more accurate solutions for complex PDEs. Similarly, “Spectral-Prior Guided Multistage Physics-Informed Neural Networks for Highly Accurate PDE Solutions” by Yuzhen Li and colleagues from the University of Electronic Science and Technology of China and Inria introduces SI-MSPINNs and RFF-MSPINNs, which use spectral information to guide network initialization, greatly improving accuracy by better learning high-energy physical modes.

Addressing the notoriously difficult problem of discontinuities, Yanpeng Gong and his team from Beijing University of Technology and Leibniz University Hannover introduce PIKAN in their paper, “Physics-Informed Kolmogorov-Arnold Networks for multi-material elasticity problems in electronic packaging”. PIKAN replaces traditional MLPs with Kolmogorov-Arnold Networks (KANs) and trainable B-spline activation functions, naturally accommodating material property discontinuities without requiring domain decomposition. This directly contrasts with the complexities highlighted by Andreas Langer and Sara Behnamian in “DeepTV: A neural network approach for total variation minimization”, which points out the theoretical issues of ReLU-NNs with discontinuous functions.

To improve training and robustness, several novel approaches emerge. Feilong Jiang and colleagues from Lancaster University and the University of Western Ontario introduce Mask-PINNs in “Mask-PINNs: Mitigating Internal Covariate Shift in Physics-Informed Neural Networks”, which uses a learnable mask to regulate feature distributions, stabilizing training and boosting accuracy. For time-dependent problems, Mayank Nagda et al. from RPTU Kaiserslautern-Landau present PIANO in “PIANO: Physics Informed Autoregressive Network”, an autoregressive framework that resolves temporal instability inherent in non-autoregressive PINNs, crucial for dynamical systems like weather forecasting. Furthermore, Gabriel Turinici from Université Paris Dauphine – PSL proposes “Regime-Aware Time Weighting for Physics-Informed Neural Networks”, using Lyapunov exponents for adaptive time weighting, enhancing convergence in chaotic and stable systems.

The challenge of obtaining robust solutions with guaranteed bounds is tackled by Liya Gaynutdinova et al. from Czech Technical University in Prague and Eindhoven University of Technology. Their paper, “Homogenization with Guaranteed Bounds via Primal-Dual Physically Informed Neural Networks”, introduces a dual formulation for PINNs, providing guaranteed upper and lower error bounds, a critical diagnostic for reliability in homogenization problems. This theoretical rigor is echoed by Ronald Katende from Kabale University and Makerere University in “Non-Asymptotic Stability and Consistency Guarantees for Physics-Informed Neural Networks via Coercive Operator Analysis”, which establishes formal stability and consistency guarantees for PINNs using coercive operator analysis.

Under the Hood: Models, Datasets, & Benchmarks

These innovations are often built upon or necessitate new architectures, datasets, and benchmarks:

• ReBaNO (Code): A lightweight neural operator leveraging knowledge distillation and reduced basis methods, demonstrating superior performance on various PDE benchmarks.
• CT-MARL (Continuous-Time Multi-Agent Reinforcement Learning): Introduced by Xuefeng Wang et al. from Purdue University in “Continuous-Time Value Iteration for Multi-Agent Reinforcement Learning”, this framework uses PINNs and Value Gradient Iteration (VGI) to solve continuous-time MARL problems, creating continuous-time versions of MPE and multi-agent MuJoCo benchmarks.
• MasconCube (Code): Developed by Pietro Fanti and Dario Izzo from ESA ESTEC for “MasconCube: Fast and Accurate Gravity Modeling with an Explicit Representation”. This self-supervised method uses a regular 3D grid of point masses for gravity modeling of small celestial bodies, achieving significant speed-ups over GeodesyNets and PINN-GM III.
• IP-Basis PINNs (Code): From Shalev Manor and Mohammad Kohandel at the University of Waterloo, described in “IP-Basis PINNs: Efficient Multi-Query Inverse Parameter Estimation”, this meta-learning approach uses offline-online decomposition and forward-mode automatic differentiation for efficient multi-query inverse problem solving.
• PINNMamba (Code): Proposed by Chenhui Xu et al. from the University at Buffalo, SUNY, in “Sub-Sequential Physics-Informed Learning with State Space Model”, this SSM-based framework mitigates over-smoothing and achieves state-of-the-art results on multiple PDE benchmarks.
• KKT-Hardnet+ (Code): A neural network architecture introduced by Ashfaq Iftakher et al. from Texas A&M University in “Physics-Informed Neural Networks with Hard Nonlinear Equality and Inequality Constraints” that strictly enforces nonlinear equality and inequality constraints through differentiable projection layers.
• PinnDE (Code): An open-source Python library by Jason Matthews and Alex Bihlo from Memorial University of Newfoundland, detailed in “PinnDE: Physics-Informed Neural Networks for Solving Differential Equations”, offering a user-friendly interface for PINNs and DeepONets with adaptive sampling strategies.
• EEMS-PINNs (Code): From Qinjiao Gao et al. at Zhejiang Gongshang University and Shanghai University of Finance and Economics, in “Energy-Equidistributed Moving Sampling Physics-informed Neural Networks for Solving Conservative Partial Differential Equations”, this framework uses energy-equidistributed adaptive sampling to ensure accuracy and energy conservation in long-time conservative PDE simulations.
• PIANO (Code): A physics-informed autoregressive network from Mayank Nagda et al. at RPTU Kaiserslautern-Landau for “PIANO: Physics Informed Autoregressive Network”, validated on challenging time-dependent PDE benchmarks and weather forecasting.
• Morephy-Net (Code): Introduced by Binghang Lu et al. from Purdue University and Utah State University in “An Evolutionary Multi-objective Optimization for Replica-Exchange-based Physics-informed Operator Learning Network”, this framework integrates evolutionary multi-objective optimization with replica exchange stochastic gradient Langevin dynamics for robust PDE solutions and uncertainty quantification.
• Pre-PINNs (Code): Proposed by Song, Tianchen et al. from Shanghai Jiao Tong University and Brown University in “A matrix preconditioning framework for physics-informed neural networks based on adjoint method”, this method addresses ill-conditioning in PINNs through matrix preconditioning, validated on Poisson and Navier–Stokes equations.
• PIKAN (Code): From Yanpeng Gong et al., for “Physics-Informed Kolmogorov-Arnold Networks for multi-material elasticity problems in electronic packaging”, demonstrating high accuracy in multi-material elasticity problems.
• SSBE-PINN (Code): Proposed by Chen, C. from The Hong Kong University of Science and Technology (HKUST) in “SSBE-PINN: A Sobolev Boundary Scheme Boosting Stability and Accuracy in Elliptic/Parabolic PDE Learning”, leveraging Sobolev spaces for improved H1 convergence in elliptic/parabolic PDEs.
• RAMS (Residual-based adversarial-gradient moving sample method): Introduced by Weihang Ouyang et al. from Hong Kong Polytechnic University and Yale University in “RAMS: Residual-based adversarial-gradient moving sample method for scientific machine learning in solving partial differential equations”, this adaptive sampling strategy uses adversarial gradients to maximize PDE residuals, enhancing accuracy and reducing computational costs.

Impact & The Road Ahead

The collective thrust of this research is clear: to make PINNs more accurate, stable, efficient, and broadly applicable. From MasconCube’s swift and interpretable gravity modeling for space exploration to the medical promise of DRetNet for diabetic retinopathy (although not PINN-focused, it highlights data-driven medical advancements that PINNs can augment) and “Towards Digital Twins for Optimal Radioembolization,” the practical implications are vast. PINNs are moving beyond academic benchmarks into real-world applications in engineering, environmental science, and even biomedical research, as exemplified by “Exploration of Hepatitis B Virus Infection Dynamics through Physics-Informed Deep Learning Approach” from Bikram Das et al. at Indian Institute of Technology Guwahati.

Advancements in adaptive sampling, such as the QR-DEIM-based strategies in “Adaptive Collocation Point Strategies For Physics Informed Neural Networks via the QR Discrete Empirical Interpolation Method” by Adrian Celaya et al. from Rice University, and the RAMS method, promise to make PINNs more robust to complex problem geometries and high-gradient regions. The development of frameworks like D3PINNs (“D3PINNs: A Novel Physics-Informed Neural Network Framework for Staged Solving of Time-Dependent Partial Differential Equations”) by Xun Yang et al. from Sichuan Normal University, which dynamically converts PDEs into ODEs, points toward hybrid methods that combine the best of neural networks with classical numerical techniques.

However, challenges remain, as underscored by “Limitations of Physics-Informed Neural Networks: a Study on Smart Grid Surrogation,” which calls for more robust and flexible frameworks for real-world smart grid dynamics. The ongoing theoretical work, such as “A convergence framework for energy minimisation of linear self-adjoint elliptic PDEs in nonlinear approximation spaces” by Alexandre Magueresse and Santiago Badia from Monash University, is crucial for establishing the rigorous foundations needed for widespread adoption. The future will likely see further integration of advanced optimization techniques, novel architectural designs (like KANs), and hybrid approaches that seamlessly blend data and physics, leading to increasingly reliable and powerful scientific machine learning solutions.