WeakIdent: weak formulation for identifying differential equation using narrow-fit and trimming. (English) Zbl 07679178
Summary: Data-driven identification of differential equations is an interesting but challenging problem, especially when the given data are corrupted by noise. When the governing differential equation is a linear combination of various differential terms, the identification problem can be formulated as solving a linear system, with the feature matrix consisting of linear and nonlinear terms multiplied by a coefficient vector. This product is equal to the time derivative term, and thus generates dynamical behaviors. The goal is to identify the correct terms that form the equation to capture the dynamics of the given data. We propose a general and robust framework to recover differential equations using a weak formulation with two new mechanisms, narrow-fit and trimming, for both ordinary and partial differential equations (ODEs and PDEs). The weak formulation facilitates an efficient and robust way to handle noise, and two new mechanisms, narrow-fit and trimming, improve the coefficient support and value recoveries respectively. For each sparsity level, Subspace Pursuit is utilized to find an initial set of support from the large dictionary. Then, we focus on highly dynamic regions (rows of the feature matrix), and error normalize the feature matrix in the narrow-fit step. The support is further updated via trimming the terms that contribute the least. Finally, the support set of features with the smallest Cross-Validation error is chosen as the result. A comprehensive set of numerical experiments are presented for both systems of ODEs and PDEs with various noise levels. The proposed method gives a robust recovery of the coefficients, and a significant denoising effect which can handle up to 100% noise-to-signal ratio for some equations. We compare the proposed method with several state-of-the-art algorithms for the recovery of differential equations.
MSC:
| 37Mxx | Approximation methods and numerical treatment of dynamical systems |
| 65Mxx | Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems |
| 35Qxx | Partial differential equations of mathematical physics and other areas of application |
Keywords:
partial differential equation identification; data-driven model selection; sparse recovery; subspace persuitReferences:
| [1] | Favela, L. H., The dynamical renaissance in neuroscience, Synthese, 199, 2103-2127 (2021) |
| [2] | Bongard, J.; Lipson, H., Automated reverse engineering of nonlinear dynamical systems, Proc. Natl. Acad. Sci., 104, 9943-9948 (2007) · Zbl 1155.37044 |
| [3] | Schmidt, M.; Lipson, H., Distilling free-form natural laws from experimental data, Science, 324, 81-85 (2009) |
| [4] | Nardini, J. T.; Lagergren, J. H.; Hawkins-Daarud, A.; Curtin, L.; Morris, B.; Rutter, E. M.; Swanson, K. R.; Flores, K. B., Learning equations from biological data with limited time samples, Bull. Math. Biol., 82, 1-33 (2020) · Zbl 1448.92241 |
| [5] | Baake, E.; Baake, M.; Bock, H.; Briggs, K., Fitting ordinary differential equations to chaotic data, Phys. Rev. A, 45, 5524 (1992) |
| [6] | Bär, M.; Hegger, R.; Kantz, H., Fitting partial differential equations to space-time dynamics, Phys. Rev. E, 59, 337 (1999) |
| [7] | Bock, H. G., Recent advances in parameter identification techniques for ode, (Numerical Treatment of Inverse Problems in Differential and Integral Equations (1983)), 95-121 · Zbl 0516.65067 |
| [8] | Müller, T. G.; Timmer, J., Parameter identification techniques for partial differential equations, Int. J. Bifurc. Chaos Appl. Sci. Eng., 14, 2053-2060 (2004) · Zbl 1062.35177 |
| [9] | Brunton, S. L.; Proctor, J. L.; Kutz, J. N., Discovering governing equations from data by sparse identification of nonlinear dynamical systems, Proc. Natl. Acad. Sci., 113, 3932-3937 (2016) · Zbl 1355.94013 |
| [10] | Zhang, L.; Schaeffer, H., On the convergence of the sindy algorithm, Multiscale Model. Simul., 17, 948-972 (2019) · Zbl 1437.37108 |
| [11] | Kaheman, K.; Kutz, J. N.; Brunton, S. L., Sindy-pi: a robust algorithm for parallel implicit sparse identification of nonlinear dynamics, Proc. Math. Phys. Eng. Sci., 476 (2020) · Zbl 1473.93007 |
| [12] | Rudy, S. H.; Alla, A.; Brunton, S. L.; Kutz, J. N., Data-driven identification of parametric partial differential equations, SIAM J. Appl. Dyn. Syst., 18, 643-660 (2019) · Zbl 1456.65096 |
| [13] | Loiseau, J.-C.; Noack, B. R.; Brunton, S. L., Sparse reduced-order modelling: sensor-based dynamics to full-state estimation, J. Fluid Mech., 844, 459-490 (2018) · Zbl 1461.76369 |
| [14] | Guan, Y.; Brunton, S. L.; Novosselov, I. V., Sparse nonlinear models of chaotic electroconvection, R. Soc. Open Sci., 8 (2021) |
| [15] | Champion, K. P.; Brunton, S. L.; Kutz, J. N., Discovery of nonlinear multiscale systems: sampling strategies and embeddings, SIAM J. Appl. Dyn. Syst., 18, 312-333 (2019) · Zbl 1474.65476 |
| [16] | Kang, S. H.; Liao, W.; Liu, Y., Ident: identifying differential equations with numerical time evolution, J. Sci. Comput., 87, 1-27 (2021) · Zbl 1467.65102 |
| [17] | He, Y.; Kang, S. H.; Liao, W.; Liu, H.; Liu, Y., Robust pde identification from noisy data (2020), arXiv preprint |
| [18] | Tran, G.; Ward, R., Exact recovery of chaotic systems from highly corrupted data, Multiscale Model. Simul., 15, 1108-1129 (2017) |
| [19] | Schaeffer, H.; Tran, G.; Ward, R.; Zhang, L., Extracting structured dynamical systems using sparse optimization with very few samples, Multiscale Model. Simul., 18, 1435-1461 (2020) · Zbl 1528.65035 |
| [20] | Rudy, S. H.; Brunton, S. L.; Proctor, J. L.; Kutz, J. N., Data-driven discovery of partial differential equations, Sci. Adv., 3, Article e1602614 pp. (2017) |
| [21] | Wu, K.; Xiu, D., Numerical aspects for approximating governing equations using data, J. Comput. Phys., 384, 200-221 (2019) · Zbl 1451.65008 |
| [22] | Gurevich, D.; Reinbold, P. A.K.; Grigoriev, R. O., Robust and optimal sparse regression for nonlinear pde models, Chaos, 29, 10, Article 103113 pp. (2019) · Zbl 1433.35387 |
| [23] | Messenger, D. A.; Bortz, D. M., Weak sindy for partial differential equations, J. Comput. Phys., Article 110525 pp. (2021) · Zbl 07515424 |
| [24] | Messenger, D. A.; Bortz, D. M., Weak sindy: Galerkin-based data-driven model selection, Multiscale Model. Simul., 19, 1474-1497 (2021) · Zbl 1512.65163 |
| [25] | Reinbold, P. A.; Gurevich, D. R.; Grigoriev, R. O., Using noisy or incomplete data to discover models of spatiotemporal dynamics, Phys. Rev. E, 101, Article 010203 pp. (2020) |
| [26] | Schaeffer, H., Learning partial differential equations via data discovery and sparse optimization, Proc. R. Soc. A, Math. Phys. Eng. Sci., 473, Article 20160446 pp. (2017) · Zbl 1404.35397 |
| [27] | Zhang, S.; Lin, G., Robust data-driven discovery of governing physical laws with error bars, Proc. R. Soc. A, Math. Phys. Eng. Sci., 474, Article 20180305 pp. (2018) · Zbl 1407.62267 |
| [28] | Chen, Z.; Churchill, V.; Wu, K.; Xiu, D., Deep neural network modeling of unknown partial differential equations in nodal space, J. Comput. Phys., 449, Article 110782 pp. (2022) · Zbl 07524779 |
| [29] | Qin, T.; Wu, K.; Xiu, D., Data driven governing equations approximation using deep neural networks, J. Comput. Phys., 395, 620-635 (2019) · Zbl 1455.65125 |
| [30] | Lusch, B.; Kutz, J. N.; Brunton, S. L., Deep learning for universal linear embeddings of nonlinear dynamics, Nat. Commun., 9, 1-10 (2018) |
| [31] | Raissi, M.; Karniadakis, G. E., Hidden physics models: machine learning of nonlinear partial differential equations, J. Comput. Phys., 357, 125-141 (2018) · Zbl 1381.68248 |
| [32] | Wu, K.; Xiu, D., Data-driven deep learning of partial differential equations in modal space, J. Comput. Phys., 408, Article 109307 pp. (2020) · Zbl 07505629 |
| [33] | Xu, H.; Chang, H.; Zhang, D., Dl-pde: deep-learning based data-driven discovery of partial differential equations from discrete and noisy data (2019), arXiv preprint |
| [34] | Xu, H.; Chang, H.; Zhang, D., Dlga-pde: discovery of pdes with incomplete candidate library via combination of deep learning and genetic algorithm, J. Comput. Phys., 418, Article 109584 pp. (2020) · Zbl 07506161 |
| [35] | He, Y.; Kang, S. H.; Liao, W.; Liu, H.; Liu, Y., Numerical identification of nonlocal potential in aggregation, Commun. Comput. Phys. (2022) |
| [36] | Rudy, S.; Alla, A.; Brunton, S. L.; Kutz, J. N., Data-driven identification of parametric partial differential equations, SIAM J. Appl. Dyn. Syst., 18, 643-660 (2019) · Zbl 1456.65096 |
| [37] | Chen, Z.; Wu, K.; Xiu, D., Methods to recover unknown processes in partial differential equations using data (2020), ArXiv abs · Zbl 1455.65159 |
| [38] | Dai, W.; Milenkovic, O., Subspace pursuit for compressive sensing signal reconstruction, IEEE Trans. Inf. Theory, 55, 2230-2249 (2009) · Zbl 1367.94082 |
| [39] | Björck, Å., Least squares methods, Handb. Numer. Anal., 1, 465-652 (1990) · Zbl 0875.65055 |
| [40] | Björck, Å., Error analysis of least squares algorithms, (Numerical Linear Algebra, Digital Signal Processing and Parallel Algorithms (1991), Springer), 41-73 · Zbl 0757.65047 |
| [41] | Brunton, S. L.; Proctor, J. L.; Kutz, J. N., Sparse identification of nonlinear dynamics with control (sindyc), IFAC-PapersOnLine, 49, 710-715 (2016) |
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