Discovering Scaling Exponents with Physics-Informed Müntz-Szász Networks
Axiom Research Group（Axiom研究组）；Department of Applied Mathematics and Computational Science（应用数学与计算科学系）；African Institute for Mathematical Sciences (AIMS), Research and Innovation Centre (RIC)（非洲数学科学研究所（AIMS）、研究与创新中心（RIC））；Graduate School of Engineering, The University of Tokyo（东京大学工学研究院）
Physical systems near singularities, interfaces, and critical points exhibit power-law scaling, yet standard neural networks leave the governing exponents implicit. We introduce physics-informed M"untz-Sz'asz Networks (MSN-PINN), a power-law basis network that treats scaling exponents as trainable parameters. The model outputs both the solution and its scaling structure. We prove identifiability, or unique recovery, and show that, under these conditions, the squared error between learned and true exponents scales as $O(|μ- α|^2)$. Across experiments, MSN-PINN achieves single-exponent recovery with 1--5% error under noise and sparse sampling. It recovers corner singularity exponents for the two-dimensional Laplace equation with 0.009% error, matches the classical result of Kondrat'ev (1967), and recovers forcing-induced exponents in singular Poisson problems with 0.03% and 0.05% errors. On a 40-configuration wedge benchmark, it reaches a 100% success rate with 0.022% mean error. Constraint-aware training encodes physical requirements such as boundary condition compatibility and improves accuracy by three orders of magnitude over naive training. By combining the expressiveness of neural networks with the interpretability of asymptotic analysis, MSN-PINN produces learned parameters with direct physical meaning.
Discovering Physics-Informed Neural Networks Model for Solving Partial Differential Equations through Evolutionary Computation
National Engineering Laboratory for Big Data Analysis and Applications, Peking University（北京大学大数据分析与应用国家工程实验室）；School of Mathematical Sciences, Peking University（北京大学数学科学学院）；PKU-Changsha Institute for Computing and Digital Economy（北京大学长沙计算与数字经济研究院）
The widespread use of neural networks across different scientific domains often involves constraining them to satisfy certain symmetries, conservation laws, or other domain knowledge. Such constraints are often imposed as soft penalties during model training and effectively act as domain-specific regularizers of the empirical risk loss. Physics-informed neural networks is an example of this philosophy in which the outputs of deep neural networks are constrained to approximately satisfy a given set of partial differential equations. In this work we review recent advances in scientific machine l…
Eigenvalue problems are critical to several fields of science and engineering. We expand on the method of using unsupervised neural networks for discovering eigenfunctions and eigenvalues for differential eigenvalue problems. The obtained solutions are given in an analytical and differentiable form that identically satisfies the desired boundary conditions. The network optimization is data-free and depends solely on the predictions of the neural network. We introduce two physics-informed loss functions. The first, called ortho-loss, motivates the network to discover pair-wise orthogonal eigenf…
Varying power-infeed from converter-based generation units introduces great uncertainty on system parameters such as inertia and damping. As a consequence, system operators face increasing challenges in performing dynamic security assessment and taking real-time control actions. Exploiting the widespread deployment of phasor measurement units (PMUs) and aiming at developing a fast dynamic state and parameter estimation tool, this paper investigates the performance of Physics-Informed Neural Networks (PINN) for discovering the frequency dynamics of future power systems. PINNs have the potential…
This work is concerned with discovering the governing partial differential equation (PDE) of a physical system. Existing methods have demonstrated the PDE identification from finite observations but failed to maintain satisfying results against noisy data, partly owing to suboptimal estimated derivatives and found PDE coefficients. We address the issues by introducing a noise-aware physics-informed machine learning (nPIML) framework to discover the governing PDE from data following arbitrary distributions. We propose training a couple of neural networks, namely solver and preselector, in a mul…
Avast Software（Avast软件公司）；Czech Technical University in Prague（布拉格捷克理工大学）
Anomaly detection is a method for discovering unusual and suspicious behavior. In many real-world scenarios, the examined events can be directly linked to the actions of an adversary, such as attacks on computer networks or frauds in financial operations. While the defender wants to discover such malicious behavior, the attacker seeks to accomplish their goal (e.g., exfiltrating data) while avoiding the detection. To this end, anomaly detectors have been used in a game-theoretic framework that captures these goals of a two-player competition. We extend the existing models to more realistic set…
Université de Paris（巴黎大学）；INSERM U1284（法国国家健康与医学研究院U1284）；Center for Research and Interdisciplinarity (CRI)（跨学科研究中心）
Model discovery aims at autonomously discovering differential equations underlying a dataset. Approaches based on Physics Informed Neural Networks (PINNs) have shown great promise, but a fully-differentiable model which explicitly learns the equation has remained elusive. In this paper we propose such an approach by integrating neural network-based surrogates with Sparse Bayesian Learning (SBL). This combination yields a robust model discovery algorithm, which we showcase on various datasets. We then identify a connection with multitask learning, and build on it to construct a Physics Informed…
Evolutionary Optimization of Physics-Informed Neural Networks: Evo-PINN Frontiers and Opportunities
Agency for Science, Technology and Research (A*STAR)（科技研究局）；Indian Institute of Technology (IIT) Goa（印度理工学院 Goa分校）；Nanyang Technological University（南洋理工大学）
EvoPINN: Agentic Discovery of Executable Algorithms for Physics-Informed Neural Networks
School of Advanced Interdisciplinary Sciences, University of Chinese Academy of Sciences（中国科学院大学先进交叉科学学院）；Institute of Automation, Chinese Academy of Sciences（中国科学院自动化研究所）
Institute of Theoretical and Computational Physics（理论与计算物理研究所）；University of Crete（克里特大学）；National University of Science and Technology MISiS（国立科技大学MISiS）
For an ensemble of nonlinear systems that model, for instance, molecules or photonic systems, we propose a method that finds efficiently the configuration that has prescribed transfer properties. Specifically, we use physics-informed machine-learning (PIML) techniques to find the parameters for the efficient transfer of an electron (or photon) to a targeted state in a non-linear dimer. We create a machine learning model containing two variables, $χ_D$, and $χ_A$, representing the non-linear terms in the donor and acceptor target system states. We then introduce a data-free physics-informed los…
Discovering novel materials can be greatly accelerated by iterative machine learning-informed proposal of candidates---active learning. However, standard \emph{global-scope error} metrics for model quality are not predictive of discovery performance, and can be misleading. We introduce the notion of \emph{Pareto shell-scope error} to help judge the suitability of a model for proposing material candidates. Further, through synthetic cases and a thermoelectric dataset, we probe the relation between acquisition function fidelity and active learning performance. Results suggest novel diagnostic to…
The application of deep learning toward discovery of data-driven models requires careful application of inductive biases to obtain a description of physics which is both accurate and robust. We present here a framework for discovering continuum models from high fidelity molecular simulation data. Our approach applies a neural network parameterization of governing physics in modal space, allowing a characterization of differential operators while providing structure which may be used to impose biases related to symmetry, isotropy, and conservation form. We demonstrate the effectiveness of our f…
Sandia National Laboratories（桑迪亚国家实验室）；Arizona State University（亚利桑那州立大学）
We introduce physics-informed multimodal autoencoders (PIMA) - a variational inference framework for discovering shared information in multimodal scientific datasets representative of high-throughput testing. Individual modalities are embedded into a shared latent space and fused through a product of experts formulation, enabling a Gaussian mixture prior to identify shared features. Sampling from clusters allows cross-modal generative modeling, with a mixture of expert decoder imposing inductive biases encoding prior scientific knowledge and imparting structured disentanglement of the latent s…
Harnessing data to discover the underlying governing laws or equations that describe the behavior of complex physical systems can significantly advance our modeling, simulation and understanding of such systems in various science and engineering disciplines. This work introduces a novel physics-informed deep learning framework to discover governing partial differential equations (PDEs) from scarce and noisy data for nonlinear spatiotemporal systems. In particular, this approach seamlessly integrates the strengths of deep neural networks for rich representation learning, physics embedding, auto…
Discovery of partial differential equations from highly noisy and sparse data with physics-informed information criterion
College of Engineering, Peking University（北京大学工学院）；Institute of Applied Physics and Computational Mathematics（应用物理与计算数学研究所）；Southern University of Science and Technology（南方科技大学）；Peng Cheng Laboratory（鹏城实验室）
Tel Aviv University（特拉维夫大学）；University College London（伦敦大学学院）
Discovering a meaningful symbolic expression that explains experimental data is a fundamental challenge in many scientific fields. We present a novel, open-source computational framework called Scientist-Machine Equation Detector (SciMED), which integrates scientific discipline wisdom in a scientist-in-the-loop approach, with state-of-the-art symbolic regression (SR) methods. SciMED combines a wrapper selection method, that is based on a genetic algorithm, with automatic machine learning and two levels of SR methods. We test SciMED on five configurations of a settling sphere, with and without …
Beijing University of Technology（北京工业大学）；Institute of Geophysics, China Earthquake Administration（中国地震局地球物理研究所）；Institute of Geomechanics, Chinese Academy of Geological Sciences（中国地质科学院地质力学研究所）
Ground-motion model (GMM) is the basis of many earthquake engineering studies. In this study, a novel physics-informed symbolic learner (PISL) method based on the Nest Generation Attenuation-West2 database is proposed to automatically discover mathematical equation operators as symbols. The sequential threshold ridge regression algorithm is utilized to distill a concise and interpretable explicit characterization of complex systems of ground motions. In addition to the basic variables retrieved from previous GMMs, the current PISL incorporates two a priori physical conditions, namely, distance…
The prediction of interfacial structure in two-phase flow systems is difficult and challenging. In this paper, a novel physics-informed reinforcement learning-aided framework (PIRLF) for the interfacial area transport is proposed. A Markov Decision Process that describes the bubble transport is established by assuming that the development of two-phase flow is a stochastic process with Markov property. The framework aims to capture the complexity of two-phase flow using the advantage of reinforcement learning (RL) in discovering complex patterns with the help of the physical model (Interfacial …
Physics-constrained robust learning of open-form partial differential equations from limited and noisy data
College of Engineering, Peking University（北京大学工学院）；Ningbo Institute of Digital Twin, Eastern Institute of Technology（东方理工大学宁波数字孪生研究院）；School of Environmental Science and Engineering, Southern University of Science and Technology（南方科技大学环境科学与工程学院）；Eastern Institute for Advanced Study, Eastern Institute of Technology（东方理工大学东方高等研究院）
