Learning the flux and diffusion function for degenerate convection-diffusion equations using different types of observations. (English) Zbl 1537.35418
Summary: In recent years, there has been an increasing interest in utilizing deep learning-based techniques to predict solutions to various partial differential equations. In this study, we investigate the identification of an unknown flux function and diffusion coefficient in a one-dimensional convection-diffusion equation. The diffusion function is allowed to vanish on intervals implying that solutions generally possess low regularity, i.e., are discontinuous. Therefore, solutions must be interpreted in the sense of entropy solutions which combine a weak formulation with an additional constraint (entropy condition). We explore a methodology that utilizes symbolic neural networks (S-Nets) in combination with an entropy-consistent discrete numerical scheme (ECDNS). Different types of observation data are explored. Extensive experiments in this paper demonstrate that the proposed method is a robust tool to identify the unknown flux and diffusion function. The flux and diffusion functions are restricted to be rational functions.
MSC:
| 35R30 | Inverse problems for PDEs |
| 35L65 | Hyperbolic conservation laws |
| 65N21 | Numerical methods for inverse problems for boundary value problems involving PDEs |
| 68T07 | Artificial neural networks and deep learning |
Keywords:
scalar nonlinear conservation law; degenerate convection-diffusion; weak entropy solution; symbolic multi-layer neural network; entropy consistent discrete numerical schemeReferences:
| [1] | Afif, M.; Amaziane, B., Convergence of finite volume schemes for a degenerate convection-diffusion equation arising in flow in porous media, Comput Methods Appl Mech Engrg, 191, 46, 5265-5285, 2002 · Zbl 1012.76057 · doi:10.1016/S0045-7825(02)00458-9 |
| [2] | Atkinson, K., An Introduction to Numerical Analysis, 1991, New York: Wiley, New York |
| [3] | Bezgin, DA; Schmidt, SJ; Adams, NA, A data-driven physics-informed finite-volume scheme for nonclassical undercompressive shocks, J. Comput. Phys., 437, 110324, 2021 · Zbl 07505908 · doi:10.1016/j.jcp.2021.110324 |
| [4] | Billard, L.; Diday, E., From the statistics of data to the statistics of knowledge: symbolic data analysis, J. Am. Stat. Assoc., 98, 462, 470-487, 2003 · doi:10.1198/016214503000242 |
| [5] | Bongard, J.; Lipson, H., Automated reverse engineering of nonlinear dynamical systems, Proc. Natl. Acad. Sci., 104, 24, 9943-9948, 2007 · Zbl 1155.37044 · doi:10.1073/pnas.0609476104 |
| [6] | Bouchut, F.; Guarguaglini, FR; Natalini, R., Diffusive bgk approximations for nonlinear multidimensional parabolic equations, Indiana Univ. Math. J., 49, 2, 749-282, 2000 · Zbl 0964.35011 · doi:10.1512/iumj.2000.49.1811 |
| [7] | Brandstetter, J., Worrall, D., Welling, M.: Message passing neural PDE solvers. In: The Tenth International Conference on Learning Representations, ICLR 2022, Virtual Event, April 25-29, 2022. OpenReview.net (2022) |
| [8] | Brunton, S.: Discovering governing equations from data by sparse identification of nonlinear dynamics. In: APS March Meeting Abstracts, volume 2017, pages X49-004 (2017) |
| [9] | Bustos, MC; Concha, F.; Bürger, R.; Tory, EM, Sedimentation and Thickening - Phenomenological Foundation and Mathematical Theory, 1999, Cambridge: Kluwer Academic Publishers, Cambridge · Zbl 0936.76001 · doi:10.1007/978-94-015-9327-4 |
| [10] | Carillo, J., Entropy solutions for nonlinear degenerate problems, Arch. Ration. Mech. Anal., 147, 4, 269-361, 1999 · Zbl 0935.35056 · doi:10.1007/s002050050152 |
| [11] | Champion, K.; Lusch, B.; Kutz, JN; Brunton, SL, Data-driven discovery of coordinates and governing equations, Proc. Natl. Acad. Sci., 116, 45, 22445-22451, 2019 · Zbl 1433.68396 · doi:10.1073/pnas.1906995116 |
| [12] | Chang, B., Chen, M., Haber, E., Chi, E.H.: Antisymmetricrnn: A dynamical system view on recurrent neural networks. In: International Conference on Learning Representations (2018) |
| [13] | Chen, R.T., Rubanova, Y., Bettencourt, J., Duvenaud, D.K.: Neural ordinary differential equations. Adv. Neural Inf. Process. Systems, 31, (2018) |
| [14] | Chen, Z.; Liu, Y.; Sun, H., Physics-informed learning of governing equations from scarce data, Nat. Commun., 12, 1, 6136, 2021 · doi:10.1038/s41467-021-26434-1 |
| [15] | Chen, Z., Zhang, J., Arjovsky, M., Bottou, L.: Symplectic recurrent neural networks. In: International Conference on Learning Representations (2019) |
| [16] | Cockburn, B.; Shu, CW, The local discontinuous Galerkin method for time dependent convection-diffusion systems, SIAM J. Numer. Anal., 35, 6, 2440-2463, 1998 · Zbl 0927.65118 · doi:10.1137/S0036142997316712 |
| [17] | Colombo, RM; Marson, A., A hölder continuous ode related to traffic flow, Proc. R. Soc. Edinb. Sect. A Math., 133, 4, 759-772, 2003 · Zbl 1052.34007 · doi:10.1017/S0308210500002663 |
| [18] | Cornforth, T., Lipson, H.: Symbolic regression of multiple-time-scale dynamical systems. In: Proceedings of the 14th annual conference on Genetic and evolutionary computation, pages 735-742 (2012) |
| [19] | Dafermos, CM; Dafermos, CM, Hyperbolic Conservation Laws in Continuum Physics, 2005, Berlin: Springer, Berlin · Zbl 1078.35001 · doi:10.1007/3-540-29089-3 |
| [20] | Dam, M.; Brøns, M.; Juul Rasmussen, J.; Naulin, V.; Hesthaven, JS, Sparse identification of a predator-prey system from simulation data of a convection model, Phys. Plasmas, 24, 2, 022310, 2017 · doi:10.1063/1.4977057 |
| [21] | DeBrouwer, E., Simm, J., Arany, A., Moreau, Y.: Gru-ode-bayes: Continuous modeling of sporadically-observed time series. Adv. Neural Inf. Process. Syst. 32, (2019) |
| [22] | Díaz-Adame, R.; Jerez, S.; Carrillo, H., Fast and optimal weno schemes for degenerate parabolic conservation laws, J. Sci. Comput., 90, 1, 22, 2022 · Zbl 1481.65126 · doi:10.1007/s10915-021-01689-4 |
| [23] | Diehl, S., Estimation of the batch-settling flux function for an ideal suspension from only two experiments, Chem. Eng. Sci., 62, 4589-4601, 2007 · doi:10.1016/j.ces.2007.05.025 |
| [24] | Diehl, S., Numerical identification of constitutive functions in scalar nonlinear convection-diffusion equations with application to batch sedimentation, Appl. Numer. Math., 95, 154-172, 2015 · Zbl 1320.65136 · doi:10.1016/j.apnum.2014.04.002 |
| [25] | Duong, D.L.: Inverse problems for hyperbolic conservation laws: a Bayesian approach. PhD thesis, University of Sussex (2021) |
| [26] | Evje, S.; Karlsen, KH, Viscous splitting approximation of mixed hyperbolic-parabolic convection-diffusion equations, Numer. Math., 83, 1, 107-137, 1999 · Zbl 0961.65084 · doi:10.1007/s002110050441 |
| [27] | Evje, S.; Karlsen, KH, Monotone difference approximations of BV solutions to degenerate convection-diffusion equations, SIAM J. Numer. Anal., 37, 6, 1838-1860, 2000 · Zbl 0985.65100 · doi:10.1137/S0036142998336138 |
| [28] | Evje, S.; Karlsen, KH, An error estimate for viscous approximate solutions of degenerate parabolic equations, J. Nonlinear Math. Phys., 9, 3, 262-281, 2002 · Zbl 1183.35180 · doi:10.2991/jnmp.2002.9.3.3 |
| [29] | Eymard, R.; Gallouet, T.; Herbin, R., Convergence of a finite volume scheme for nonlinear degenerate parabolic equations, Numer. Math., 92, 41-82, 2002 · Zbl 1005.65099 · doi:10.1007/s002110100342 |
| [30] | Fuks, O., Tchelepi, H.A.: Limitations of physics informed machine learning for nonlinear two-phase transport in porous media. J. Mach. Learn. Model. Comput., 1(1) (2020) |
| [31] | Galiano, SJ; Zapata, MU, A new tvd flux-limiter method for solving nonlinear hyperbolic equations, J. Comput. Appl. Math., 234, 5, 1395-1403, 2010 · Zbl 1205.65234 · doi:10.1016/j.cam.2010.02.015 |
| [32] | Gao, H.; Sun, L.; Wang, JX, Phygeonet: Physics-informed geometry-adaptive convolutional neural networks for solving parameterized steady-state pdes on irregular domain, J. Comput. Phys., 428, 110079, 2021 · Zbl 07511433 · doi:10.1016/j.jcp.2020.110079 |
| [33] | Gao, H.; Zahr, MJ; Wang, JX, Physics-informed graph neural galerkin networks: A unified framework for solving pde-governed forward and inverse problems, Comput. Methods Appl. Mech. Eng., 390, 114502, 2022 · Zbl 1507.65223 · doi:10.1016/j.cma.2021.114502 |
| [34] | Gaucel, S., Keijzer, M., Lutton, E., Tonda, A.: Learning dynamical systems using standard symbolic regression. In: Genetic Programming: 17th European Conference, EuroGP 2014, Granada, Spain, April 23-25, 2014, Revised Selected Papers 17, pp. 25-36. Springer (2014) |
| [35] | Herrera, C., Krach, F., Teichmann, J.: Neural jump ordinary differential equations: Consistent continuous-time prediction and filtering. In: International Conference on Learning Representations (2020) |
| [36] | Hesthaven, J.S.: Numerical methods for conservation laws: From analysis to algorithms. SIAM. Comput. Sci. Eng. (2017) |
| [37] | Holden, H., Karlsen, K.H., Lie, K.A., Risebro, N.H.: Splitting methods for partial differential equations with rough solutions. Eur. Math. Soc. (2010) · Zbl 1191.35005 |
| [38] | Holden, H.; Priuli, FS; Risebro, NH, On an inverse problem for scalar conservation laws, Inverse Prob., 30, 035015, 2014 · Zbl 1286.35266 · doi:10.1088/0266-5611/30/3/035015 |
| [39] | Iakovlev, V., Heinonen, M., Lähdesmäki, H.: Learning continuous-time pdes from sparse data with graph neural networks. In: 9th International Conference on Learning Representations, ICLR 2021, Virtual Event, Austria, May 3-7, 2021. OpenReview.net (2021) |
| [40] | James, F.; Sepúlveda, M., Parameter identification for a model of chromatographic column, Inverse Prob., 10, 6, 1299, 1994 · Zbl 0809.35159 · doi:10.1088/0266-5611/10/6/008 |
| [41] | James, F.; Sepúlveda, M., Convergence results for the flux identification in a scalar conservation law, SIAM J. Control. Optim., 37, 3, 869-891, 1999 · Zbl 0970.35161 · doi:10.1137/S0363012996272722 |
| [42] | James, F.; Sepúlveda, M., Convergence results for the flux identification in a scalar conservation law, SIAM J. Control. Optim., 37, 3, 869-891, 1999 · Zbl 0970.35161 · doi:10.1137/S0363012996272722 |
| [43] | Kaheman, K.; Kutz, JN; Brunton, SL, Sindy-pi: a robust algorithm for parallel implicit sparse identification of nonlinear dynamics, Proc. R. Soc. A, 476, 2242, 20200279, 2020 · Zbl 1473.93007 · doi:10.1098/rspa.2020.0279 |
| [44] | Kang, H.; Tanuma, K., Inverse problems for scalar conservation laws, Inverse Prob., 21, 3, 1047, 2005 · Zbl 1071.35125 · doi:10.1088/0266-5611/21/3/015 |
| [45] | Karlsen, KH; Risebro, NH; Storrøsten, EB, \(L^1\) error estimates for difference approximations of degenerate convection-diffusion equations, Math. Comp., 83, 290, 2717-2762, 2014 · Zbl 1300.65067 · doi:10.1090/S0025-5718-2014-02818-4 |
| [46] | Karlsen, KH; Risebro, NH; Storrøsten, EB, On the convergence rate of finite difference methods for degenerate convection-diffusion equations in several space dimensions, ESAIM Math. Modell. Numer. Anal., 50, 2, 499-539, 2016 · Zbl 1342.65182 · doi:10.1051/m2an/2015057 |
| [47] | Kim, S.; Lu, PY; Mukherjee, S.; Gilbert, M.; Jing, L.; Čeperić, V.; Soljačić, M., Integration of neural network-based symbolic regression in deep learning for scientific discovery, IEEE Trans. Neural Netw. Learn. Syst., 32, 9, 4166-4177, 2020 · doi:10.1109/TNNLS.2020.3017010 |
| [48] | Koley, U.; Risebro, NH; Schwab, C.; Weber, F., A multilevel Monte Carlo finite difference method for random scalar degenerate convection diffusion equations, J. Hyperbolic Differ. Equ., 14, 3, 415-445, 2017 · Zbl 1373.35179 · doi:10.1142/S021989161750014X |
| [49] | Kröener, D.: Numerical schemes for conservation laws. Wiley-Teubner Series Advances in Numerical Mathematics (1997) · Zbl 0872.76001 |
| [50] | Kruzkov, SN, First order quasilinear equations with several independent variables, Mat. Sb. (N.S.), 81, 123, 228-255, 1970 · Zbl 0202.11203 |
| [51] | Kurganov, A.; Tadmor, E., New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations, J. Comput. Phys., 160, 1, 241-282, 2000 · Zbl 0987.65085 · doi:10.1006/jcph.2000.6459 |
| [52] | LeVeque, R.J.: Finite volume methods for hyperbolic problems. Cambridge Texts in Applied Mathematics (2007) |
| [53] | Li, Q., Evje, S.: Learning the nonlinear flux function of a hidden scalar conservation law from data. Network Heterogeneous Media, 18, (2023) · Zbl 1531.68086 |
| [54] | Li, Q.; Evje, S.; Geng, J., Learning parameterized odes from data, IEEE Access, 11, 54897-54909, 2023 · doi:10.1109/ACCESS.2023.3282435 |
| [55] | Li, Q.; Geng, J.; Evje, S., Identification of the flux function of nonlinear conservation laws with variable parameters, Physica D, 451, 133773, 2023 · Zbl 1517.35263 · doi:10.1016/j.physd.2023.133773 |
| [56] | Li, Q.; Geng, J.; Evje, S.; Rong, C., Solving nonlinear conservation laws of partial differential equations using graph neural networks, Proc. Northern Lights Deep Learn. Workshop, 2023, 4, 2023 |
| [57] | Li, Q.; Evje, S., Learning the nonlinear flux function of a hidden scalar conservation law from data, Netw. Heterogen. Media, 18, 1, 48-79, 2023 · Zbl 1531.68086 · doi:10.3934/nhm.2023003 |
| [58] | Lighthill, MJ; Whitham, GB, On kinematic waves ii. a theory of traffic flow on long crowded roads, Proc. R. Soc. Lond. A, 229, 1178, 317-345, 1955 · Zbl 0064.20906 · doi:10.1098/rspa.1955.0089 |
| [59] | Liu, XD; Oshery, S.; Chanz, T., Weighted essentially non-oscillatory schemes, J. Comput. Phys., 115, 1, 200-212, 1994 · Zbl 0811.65076 · doi:10.1006/jcph.1994.1187 |
| [60] | Liu, Z.; Madhavan, V.; Tegmark, M., Machine learning conservation laws from differential equations, Phys. Rev. E, 106, 4, 045307, 2022 · doi:10.1103/PhysRevE.106.045307 |
| [61] | Liu, Z.; Tegmark, M., Machine learning conservation laws from trajectories, Phys. Rev. Lett., 126, 18, 180604, 2021 · doi:10.1103/PhysRevLett.126.180604 |
| [62] | Long, Z.; Lu, Y.; Dong, B., PDE-Net 2.0: Learning PDEs from data with a numeric-symbolic hybrid deep network, J. Comput. Phys., 399, 108925, 2019 · Zbl 1454.65131 · doi:10.1016/j.jcp.2019.108925 |
| [63] | Long, Z., Lu, Y., Ma, X., Dong, B.: Pde-net: Learning PDEs from data. In: International Conference on Machine Learning, pages 3208-3216. PMLR (2018) |
| [64] | Magiera, J.; Ray, D.; Hesthaven, JS; Rohde, C., Constraint-aware neural networks for riemann problems, J. Comput. Phys., 409, 109345, 2020 · Zbl 1435.76046 · doi:10.1016/j.jcp.2020.109345 |
| [65] | Martius, G., Lampert, C.H.: Extrapolation and learning equations. In: 5th International Conference on Learning Representations, ICLR 2017, Toulon, France, April 24-26, 2017, Workshop Track Proceedings. OpenReview.net (2017) |
| [66] | Mishra, S.; Schwab, C., Sparse tensor multi-level monte carlo finite volume methods for hyperbolic conservation laws with random initial data, Math. Comput., 81, 280, 1979-2018, 2012 · Zbl 1271.65018 · doi:10.1090/S0025-5718-2012-02574-9 |
| [67] | Mototake, YI, Interpretable conservation law estimation by deriving the symmetries of dynamics from trained deep neural networks, Phys. Rev. E, 103, 3, 033303, 2021 · doi:10.1103/PhysRevE.103.033303 |
| [68] | Mundhenk, T.N., Landajuela, M., Glatt, R., Santiago, C.P., Faissol, D.M., Petersen, B.K.: Symbolic regression via neural-guided genetic programming population seeding. arXiv preprint arXiv:2111.00053 (2021) |
| [69] | Narasingam, A.; Kwon, JSI, Data-driven identification of interpretable reduced-order models using sparse regression, Comput. Chem. Eng., 119, 101-111, 2018 · doi:10.1016/j.compchemeng.2018.08.010 |
| [70] | Ohlberger, M., A posteriori error estimates for vertex centered finite volume approximations of convection-diffusion-reaction equations, M2AN Math. Model Numer. Anal., 35, 2, 355-387, 2002 · Zbl 0992.65100 · doi:10.1051/m2an:2001119 |
| [71] | Petersen, B.K., Landajuela, M., Mundhenk, T.N., Santiago, C.P., Kim, S.K., Kim, J.T.: Deep symbolic regression: Recovering mathematical expressions from data via risk-seeking policy gradients. arXiv preprint arXiv:1912.04871 (2019) |
| [72] | Pfaff, T., Fortunato, M., Sanchez-Gonzalez, A., Battaglia, P.W.: Learning mesh-based simulation with graph networks. In: 9th International Conference on Learning Representations, ICLR 2021, Virtual Event, Austria, May 3-7, 2021. OpenReview.net (2021) |
| [73] | Raissi, M.; Perdikaris, P.; Karniadakis, GE, Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, J. Comput. Phys., 378, 686-707, 2019 · Zbl 1415.68175 · doi:10.1016/j.jcp.2018.10.045 |
| [74] | Ray, D.; Hesthaven, JS, An artificial neural network as a troubled-cell indicator, J. Comput. Phys., 367, 166-191, 2018 · Zbl 1415.65229 · doi:10.1016/j.jcp.2018.04.029 |
| [75] | Richards, PI, Shock waves on the highway, Oper. Res., 4, 1, 42-51, 1956 · Zbl 1414.90094 · doi:10.1287/opre.4.1.42 |
| [76] | Rubanova, Y., Chen, R.T., Duvenaud, D.K.: Latent ordinary differential equations for irregularly-sampled time series. Adv. Neural Inf. Process. Syst. 32, (2019) |
| [77] | Ruder, S.: An overview of gradient descent optimization algorithms. arXiv preprint arXiv:1609.04747 (2016) |
| [78] | Sahoo, S., Lampert, C., Martius, G.: Learning equations for extrapolation and control. In: International Conference on Machine Learning, pp. 4442-4450. PMLR (2018) |
| [79] | Schaeffer, H., Learning partial differential equations via data discovery and sparse optimization, Proc. R. Soc. A Math. Phys. Eng. Sci., 473, 2197, 20160446, 2017 · Zbl 1404.35397 |
| [80] | Schmidt, M.; Lipson, H., Distilling free-form natural laws from experimental data, Science, 324, 5923, 81-85, 2009 · doi:10.1126/science.1165893 |
| [81] | Shu, C.W.: Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. Adv. Numerical Approximation of Nonlinear Hyperbolic Equ., pp. 325-432 (1998) · Zbl 0927.65111 |
| [82] | Skadsem, HJ; Kragset, S., A numerical study of density-unstable reverse circulation displacement for primary cementing, J. Energy Res. Technol., 144, 123008, 2022 · doi:10.1115/1.4054367 |
| [83] | Thuerey, N.; Weißenow, K.; Prantl, L.; Hu, X., Deep learning methods for reynolds-averaged navier-stokes simulations of airfoil flows, AIAA J., 58, 1, 25-36, 2020 · doi:10.2514/1.J058291 |
| [84] | Vaddireddy, H.; Rasheed, A.; Staples, AE; San, O., Feature engineering and symbolic regression methods for detecting hidden physics from sparse sensor observation data, Phys. Fluids, 32, 1, 015113, 2020 · doi:10.1063/1.5136351 |
| [85] | Volpert, AI, Generalized solutions of degenerate second-order quasilinear parabolic and elliptic equations, Adv. Differ. Equ., 5, 10-12, 1493-1518, 2000 · Zbl 0988.35091 |
| [86] | Volpert, AI; Hudjaev, SI, The cauchy problem for second order quasilinear degenerate parabolic equations, Mat. Sb. (N.S.), 78, 120, 374-396, 1969 · Zbl 0181.37601 |
| [87] | Wandel, N., Weinmann, M., Klein, R.: Learning incompressible fluid dynamics from scratch - towards fast, differentiable fluid models that generalize. In: 9th International Conference on Learning Representations, ICLR 2021, Virtual Event, Austria, May 3-7, 2021. OpenReview.net (2021) |
| [88] | Wang, Y., Shen, Z., Long, Z., Dong, B.: Learning to discretize: solving 1d scalar conservation laws via deep reinforcement learning. arXiv preprint arXiv:1905.11079 (2019) |
| [89] | Zhao, Q., Lindell, D.B., Wetzstein, G.: Learning to solve pde-constrained inverse problems with graph networks. In: Kamalika, C., Stefanie, J., Le, S., Csaba, S., Gang, N., Sivan, S. (eds.), International Conference on Machine Learning, ICML 2022, 17-23 July 2022, Baltimore, Maryland, USA, volume 162 of Proceedings of Machine Learning Research, pp. 26895-26910. PMLR (2022) |
| [90] | Zhu, C.; Byrd, RH; Lu, P.; Nocedal, J., Algorithm 778: L-bfgs-b: Fortran subroutines for large-scale bound-constrained optimization, ACM Trans. Math. Softw. (TOMS), 23, 4, 550-560, 1997 · Zbl 0912.65057 · doi:10.1145/279232.279236 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
