An official website of the United States government
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.
more »
« less
Data-driven discovery of chemotactic migration of bacteria via coordinate-invariant machine learning
Abstract BackgroundE. colichemotactic motion in the presence of a chemonutrient field can be studied using wet laboratory experiments or macroscale-level partial differential equations (PDEs) (among others). Bridging experimental measurements and chemotactic Partial Differential Equations requires knowledge of the evolution of all underlying fields, initial and boundary conditions, and often necessitates strong assumptions. In this work, we propose machine learning approaches, along with ideas from the Whitney and Takens embedding theorems, to circumvent these challenges. ResultsMachine learning approaches for identifying underlying PDEs were (a) validated through the use of simulation data from established continuum models and (b) used to infer chemotactic PDEs from experimental data. Such data-driven models were surrogates either for the entire chemotactic PDE right-hand-side (black box models), or, in a more targeted fashion, just for the chemotactic term (gray box models). Furthermore, it was demonstrated that a short history of bacterial density may compensate for the missing measurements of the field of chemonutrient concentration. In fact, given reasonable conditions, such a short history of bacterial density measurements could even be used toinferchemonutrient concentration. ConclusionData-driven PDEs are an important modeling tool when studying Chemotaxis at the macroscale, as they can learn bacterial motility from various data sources, fidelities (here, computational models, experiments) or coordinate systems. The resulting data-driven PDEs can then be simulated to reproduce/predict computational or experimental bacterial density profile data independent of the coordinate system, approximate meaningful parameters or functional terms, and even possibly estimate the underlying (unmeasured) chemonutrient field evolution.
more »
« less
- Award ID(s):
- 1941716
- PAR ID:
- 10550461
- Publisher / Repository:
- Springer Science + Business Media
- Date Published:
- Journal Name:
- BMC Bioinformatics
- Volume:
- 25
- Issue:
- 1
- ISSN:
- 1471-2105
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
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.more » « less
-
IntroductionThe moment quantities associated with the nonlinear Schrödinger equation offer important insights into the evolution dynamics of such dispersive wave partial differential equation (PDE) models. The effective dynamics of the moment quantities are amenable to both analytical and numerical treatments. MethodsIn this paper, we present a data-driven approach associated with the “Sparse Identification of Nonlinear Dynamics” (SINDy) to capture the evolution behaviors of such moment quantities numerically. Results and DiscussionOur method is applied first to some well-known closed systems of ordinary differential equations (ODEs) which describe the evolution dynamics of relevant moment quantities. Our examples are, progressively, of increasing complexity and our findings explore different choices within the SINDy library. We also consider the potential discovery of coordinate transformations that lead to moment system closure. Finally, we extend considerations to settings where a closed analytical form of the moment dynamics is not available.more » « less
-
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.more » « less
-
Recent advances in high-resolution imaging techniques and particle-based simulation methods have enabled the precise microscopic characterization of collective dynamics in various biological and engineered active matter systems. In parallel, data-driven algorithms for learning interpretable continuum models have shown promising potential for the recovery of underlying partial differential equations (PDEs) from continuum simulation data. By contrast, learning macroscopic hydrodynamic equations for active matter directly from experiments or particle simulations remains a major challenge, especially when continuum models are not known a priori or analytic coarse graining fails, as often is the case for nondilute and heterogeneous systems. Here, we present a framework that leverages spectral basis representations and sparse regression algorithms to discover PDE models from microscopic simulation and experimental data, while incorporating the relevant physical symmetries. We illustrate the practical potential through a range of applications, from a chiral active particle model mimicking nonidentical swimming cells to recent microroller experiments and schooling fish. In all these cases, our scheme learns hydrodynamic equations that reproduce the self-organized collective dynamics observed in the simulations and experiments. This inference framework makes it possible to measure a large number of hydrodynamic parameters in parallel and directly from video data.more » « less
