The DOI matches its Crossref record; 41 of the 41 references it cites were found in a registry or an index.
PDE-Net 2.0: Learning PDEs from data with a numeric-symbolic hybrid deep network
Open accessAbstract
Abstract Partial differential equations (PDEs) are commonly derived based on empirical observations. However, recent advances of technology enable us to collect and store massive amount of data, which offers new opportunities for data-driven discovery of PDEs. In this paper, we propose a new deep neural network, called PDE-Net 2.0, to discover (time-dependent) PDEs from observed dynamic data with minor prior knowledge on the underlying mechanism that drives the dynamics. The design of PDE-Net 2.0 is based on our earlier work [1] where the original version of PDE-Net was proposed. PDE-Net 2.0 is a combination of numerical approximation of differential operators by convolutions and a symbolic multi-layer neural network for model recovery. Comparing with existing approaches, PDE-Net 2.0 has the most flexibility and expressive power by learning both differential operators and the nonlinear response function of the underlying PDE model. Numerical experiments show that the PDE-Net 2.0 has the potential to uncover the hidden PDE of the observed dynamics, and predict the dynamical behavior for a relatively long time, even in a noisy environment.
Overview
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Integrity signals
Registry
Cites
Statistics
Verification
Claims the paper makes
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
The self-check needs the paper’s full text.
Citations
Cites
Deep Residual Learning for Image Recognition
He, K., Zhang, X., Ren, S., & Sun, J. (2016) · 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR) · doi:10.1109/cvpr.2016.90 · Cited by 235,474
Proceedings of IEEE Conference on Computer Vision and Pattern Recognition
(1993) · doi:10.1109/cvpr.1993 · Cited by 17,667
A Wavelet Tour of Signal Processing
(1999) · Elsevier eBooks · doi:10.1016/b978-0-12-466606-1.x5000-4 · Cited by 16,227
Identity mappings in deep residual networks
He, K., Zhang, X., Ren, S., & Sun, J. (2016) · Lecture Notes in Computer Science · doi:10.1007/978-3-319-46493-0_38 · Cited by 10,741
Identity Mappings in Deep Residual Networks
He, K., Zhang, X., Ren, S., & Sun, J. (2016) · arXiv · doi:10.48550/arxiv.1603.05027 · Cited by 10,665
Finite Volume Methods for Hyperbolic Problems
LeVeque, R. J. (2002) · Cambridge University Press eBooks · doi:10.1017/cbo9780511791253 · Cited by 6,313
Discovering governing equations from data by sparse identification of nonlinear dynamical systems
Brunton, S. L., Proctor, J. L., & Kutz, J. N. (2016) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.1517384113 · Cited by 4,796
Weighted Essentially Non-oscillatory Schemes
Liu, X.-D., Osher, S., & Chan, T. (1994) · Journal of Computational Physics · doi:10.1006/jcph.1994.1187 · Cited by 3,437
Distilling Free-Form Natural Laws from Experimental Data
Schmidt, M., & Lipson, H. (2009) · Science · doi:10.1126/science.1165893 · Cited by 2,890
Deep Potential Molecular Dynamics: A Scalable Model with the Accuracy of Quantum Mechanics
Zhang, L., Han, J., Wang, H., Car, R., & E, W. (2018) · Physical Review Letters · doi:10.1103/physrevlett.120.143001 · Cited by 2,258
Cited by
Cheng, S., Quilodrán-Casas, C., Ouala, S., Farchi, A., Liu, C., Tandeo, P., Fablet, R., Lucor, D., Iooss, B., Brajard, J., et al. (2023) · IEEE/CAA Journal of Automatica Sinica · doi:10.1109/jas.2023.123537 · Cited by 303
Integration of Neural Network-Based Symbolic Regression in Deep Learning for Scientific Discovery
Kim, S., Lu, P. Y., Mukherjee, S., Gilbert, M., Jing, L., Ceperic, V., & Soljacic, M. (2021) · IEEE Transactions on Neural Networks and Learning Systems · doi:10.1109/tnnls.2020.3017010 · Cited by 215
Myung, C. W., Hajibabaei, A., Cha, J., Ha, M., Kim, J., & Kim, K. S. (2022) · Advanced Energy Materials · doi:10.1002/aenm.202202279 · Cited by 54
Metadata
- Authors
- Long, Z., Lu, Y., & Dong, B.
- Venue
- Journal of Computational Physics
- Publisher
- Elsevier BV
- Published
- 2019-12
- Volume
- 399
- Pages
- 108925
- ISSN
- 0021-9991 (ISSN-L)
- OpenAlex
- W2903660960
- Cited by
- 549
- References
- 41
- FWCI
- 25.68
- Type
- journal article
- Language
- en
- Open access
- green
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
Are you an author, or do you see a problem with this page? Request a review
How to cite
Cite CoChat for the checks on this page, and the paper for its own findings.
- This page
- CoChat. (2026). Citation check: PDE-Net 2.0: Learning PDEs from data with a numeric-symbolic hybrid deep network. https://app.cochat.ai/papers/10.1016/j.jcp.2019.108925
- The paper
- Long, Z., Lu, Y., & Dong, B. (2019). PDE-Net 2.0: Learning PDEs from data with a numeric-symbolic hybrid deep network. Journal of Computational Physics. https://doi.org/10.1016/j.jcp.2019.108925
Data sources
Bibliographic data: Semantic Scholar, OpenAlex, Crossref.
Contains information from the Semantic Scholar Academic Graph Datasets, made available under the ODC Attribution License (ODC-BY 1.0).
Semantic Scholar’s data is described in: Kinney, R., et al. (2023). The Semantic Scholar Open Data Platform. arXiv:2301.10140.
