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1. Multi-resolution partial differential equations preserved learning framework for spatiotemporal dynamics

   https://doi.org/10.1038/s42005-024-01521-z  

   Liu, Xin-Yang; Zhu, Min; Lu, Lu; Sun, Hao; Wang, Jian-Xun (January 2024, Communications Physics)

   Abstract Traditional data-driven deep learning models often struggle with high training costs, error accumulation, and poor generalizability in complex physical processes. Physics-informed deep learning (PiDL) addresses these challenges by incorporating physical principles into the model. Most PiDL approaches regularize training by embedding governing equations into the loss function, yet this depends heavily on extensive hyperparameter tuning to weigh each loss term. To this end, we propose to leverage physics prior knowledge by “baking” the discretized governing equations into the neural network architecture via the connection between the partial differential equations (PDE) operators and network structures, resulting in a PDE-preserved neural network (PPNN). This method, embedding discretized PDEs through convolutional residual networks in a multi-resolution setting, largely improves the generalizability and long-term prediction accuracy, outperforming conventional black-box models. The effectiveness and merit of the proposed methods have been demonstrated across various spatiotemporal dynamical systems governed by spatiotemporal PDEs, including reaction-diffusion, Burgers’, and Navier-Stokes equations. 

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2. Learning the governing partial differential equation of solid diffusion from atomistic simulations

   https://doi.org/10.1098/rspa.2025.0383  

   Nadew, Wongelemengist; Wang, Haoran (September 2025, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences)

   Governing partial differential equations (PDEs) play a critical role in materials research and applications, as they describe essential physics underlying materials behaviour. Traditionally, these equations are developed through phenomenological modelling of experimental results or first principle analysis based on conservation laws. In addition, molecular dynamics (MD) simulations capture atomistic-scale behaviour with detailed physics. However, translating atomistic insights into continuum-scale governing equations remains a significant challenge. Empowered by recent advances in data-driven modelling, we develop a computational framework to learn governing PDEs directly from atomistic simulation data. The framework integrates numerical differentiation of MD data with the identification of constitutive relationships. It proves effective and efficient in learning governing PDEs from noisy and limited MD datasets, without requiring prior knowledge of the final PDEs. Using this framework, we identify a nonlinear PDE governing solid-state diffusion in nickel–hydrogen alloys. This PDE reveals a highly concentration-dependent diffusivity that varies over an order of magnitude. Our data-driven computational framework paves the way for cross-scale constitutive modelling. 

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3. Physics-Informed Neural Network-Based Inverse Framework for Time-Fractional Differential Equations for Rheology

   https://doi.org/10.3390/biology14070779  

   Thakur, Sukirt; Mitra, Harsa; Ardekani, Arezoo M (July 2025, Biology)

   Inverse problems involving time-fractional differential equations have become increasingly important for modeling systems with memory-dependent dynamics, particularly in biotransport and viscoelastic materials. Despite their potential, these problems remain challenging due to issues of stability, non-uniqueness, and limited data availability. Recent advancements in Physics-Informed Neural Networks (PINNs) offer a data-efficient framework for solving such inverse problems, yet most implementations are restricted to integer-order derivatives. In this work, we develop a PINN-based framework tailored for inverse problems involving time-fractional derivatives. We consider two representative applications: anomalous diffusion and fractional viscoelasticity. Using both synthetic and experimental datasets, we infer key physical parameters including the generalized diffusion coefficient and the fractional derivative order in the diffusion model and the relaxation parameters in a fractional Maxwell model. Our approach incorporates a customized residual loss function scaled by the standard deviation of observed data to enhance robustness. Even under 25% Gaussian noise, our method recovers model parameters with relative errors below 10%. Additionally, the framework accurately predicts relaxation moduli in porcine tissue experiments, achieving similar error margins. These results demonstrate the framework’s effectiveness in learning fractional dynamics from noisy and sparse data, paving the way for broader applications in complex biological and mechanical systems. 

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4. White-box Machine learning approaches to identify governing equations for overall dynamics of manufacturing systems: A case study on distillation column

   Subramanian, Renganathan; Moar, Raghav Rajesh; Singh, Shweta (March 2021, Machine learning with applications)

   null (Ed.)

   Dynamical equations form the basis of design for manufacturing processes and control systems; however, identifying governing equations using a mechanistic approach is tedious. Recently, Machine learning (ML) has shown promise to identify the governing dynamical equations for physical systems faster. This possibility of rapid identification of governing equations provides an exciting opportunity for advancing dynamical systems modeling. However, applicability of the ML approach in identifying governing mechanisms for the dynamics of complex systems relevant to manufacturing has not been tested. We test and compare the efficacy of two white-box ML approaches (SINDy and SymReg) for predicting dynamics and structure of dynamical equations for overall dynamics in a distillation column. Results demonstrate that a combination of ML approaches should be used to identify a full range of equations. In terms of physical law, few terms were interpretable as related to Fick’s law of diffusion and Henry’s law in SINDy, whereas SymReg identified energy balance as driving dynamics. 

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5. Robust data-driven discovery of governing physical laws with error bars

   https://doi.org/10.1098/rspa.2018.0305  

   Zhang, Sheng; Lin, Guang (September 2018, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences)

   Discovering governing physical laws from noisy data is a grand challenge in many science and engineering research areas. We present a new approach to data-driven discovery of ordinary differential equations (ODEs) and partial differential equations (PDEs), in explicit or implicit form. We demonstrate our approach on a wide range of problems, including shallow water equations and Navier–Stokes equations. The key idea is to select candidate terms for the underlying equations using dimensional analysis, and to approximate the weights of the terms with error bars using our threshold sparse Bayesian regression. This new algorithm employs Bayesian inference to tune the hyperparameters automatically. Our approach is effective, robust and able to quantify uncertainties by providing an error bar for each discovered candidate equation. The effectiveness of our algorithm is demonstrated through a collection of classical ODEs and PDEs. Numerical experiments demonstrate the robustness of our algorithm with respect to noisy data and its ability to discover various candidate equations with error bars that represent the quantified uncertainties. Detailed comparisons with the sequential threshold least-squares algorithm and the lasso algorithm are studied from noisy time-series measurements and indicate that the proposed method provides more robust and accurate results. In addition, the data-driven prediction of dynamics with error bars using discovered governing physical laws is more accurate and robust than classical polynomial regressions. 

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