[{"status":"public","abstract":[{"lang":"eng","text":"We utilize extreme learning machines for the prediction of partial differential equations (PDEs). Our method splits the state space into multiple windows that are predicted individually using a single model. Despite requiring only few data points (in some cases, our method can learn from a single full-state snapshot), it still achieves high accuracy and can predict the flow of PDEs over long time horizons. Moreover, we show how additional symmetries can be exploited to increase sample efficiency and to enforce equivariance."}],"type":"preprint","language":[{"iso":"eng"}],"keyword":["extreme learning machines","partial differential equations","data-driven prediction","high-dimensional systems"],"user_id":"98879","_id":"53793","citation":{"apa":"Harder, H., &#38; Peitz, S. (n.d.). <i>Predicting PDEs Fast and Efficiently with Equivariant Extreme Learning Machines</i>.","short":"H. Harder, S. Peitz, (n.d.).","mla":"Harder, Hans, and Sebastian Peitz. <i>Predicting PDEs Fast and Efficiently with Equivariant Extreme Learning Machines</i>.","bibtex":"@article{Harder_Peitz, title={Predicting PDEs Fast and Efficiently with Equivariant Extreme Learning Machines}, author={Harder, Hans and Peitz, Sebastian} }","ama":"Harder H, Peitz S. Predicting PDEs Fast and Efficiently with Equivariant Extreme Learning Machines.","ieee":"H. Harder and S. Peitz, “Predicting PDEs Fast and Efficiently with Equivariant Extreme Learning Machines.” .","chicago":"Harder, Hans, and Sebastian Peitz. “Predicting PDEs Fast and Efficiently with Equivariant Extreme Learning Machines,” n.d."},"year":"2024","publication_status":"unpublished","main_file_link":[{"url":"https://arxiv.org/abs/2404.18530","open_access":"1"}],"title":"Predicting PDEs Fast and Efficiently with Equivariant Extreme Learning Machines","date_created":"2024-04-30T08:43:14Z","author":[{"full_name":"Harder, Hans","id":"98879","last_name":"Harder","first_name":"Hans"},{"last_name":"Peitz","orcid":"0000-0002-3389-793X","id":"47427","full_name":"Peitz, Sebastian","first_name":"Sebastian"}],"oa":"1","date_updated":"2024-04-30T08:45:24Z"},{"file":[{"relation":"main_file","content_type":"application/pdf","access_level":"open_access","file_id":"32801","file_name":"BEA_MultiStep_Matrix.pdf","description":"The numerical solution of an ordinary differential equation can be interpreted as the exact solution of a nearby modified equation. Investigating the behaviour of numerical solutions by analysing the modified equation is known as backward error analysis. If the original and modified equation share structural properties, then the exact and approximate solution share geometric features such as the existence of conserved quantities. Conjugate symplectic methods preserve a modified symplectic form and a modified Hamiltonian when applied to a Hamiltonian system. We show how a blended version of variational and symplectic techniques can be used to compute modified symplectic and Hamiltonian structures. In contrast to other approaches, our backward error analysis method does not rely on an ansatz but computes the structures systematically, provided that a variational formulation of the method is known. The technique is illustrated on the example of symmetric linear multistep methods with matrix coefficients.","file_size":827030,"title":"Backward error analysis for conjugate symplectic methods","date_created":"2022-08-12T16:48:59Z","creator":"coffen","date_updated":"2022-08-12T16:48:59Z"}],"abstract":[{"lang":"eng","text":"The numerical solution of an ordinary differential equation can be interpreted as the exact solution of a nearby modified equation. Investigating the behaviour of numerical solutions by analysing the modified equation is known as backward error analysis. If the original and modified equation share structural properties, then the exact and approximate solution share geometric features such as the existence of conserved quantities. Conjugate symplectic methods preserve a modified symplectic form and a modified Hamiltonian when applied to a Hamiltonian system. We show how a blended version of variational and symplectic techniques can be used to compute modified symplectic and Hamiltonian structures. In contrast to other approaches, our backward error analysis method does not rely on an ansatz but computes the structures systematically, provided that a variational formulation of the method is known. The technique is illustrated on the example of symmetric linear multistep methods with matrix coefficients."}],"publication":"Journal of Geometric Mechanics","language":[{"iso":"eng"}],"ddc":["510"],"keyword":["variational integrators","backward error analysis","Euler--Lagrange equations","multistep methods","conjugate symplectic methods"],"external_id":{"arxiv":["2201.03911"]},"year":"2023","issue":"1","quality_controlled":"1","title":"Backward error analysis for conjugate symplectic methods","date_created":"2022-01-11T12:48:39Z","publisher":"AIMS Press","status":"public","type":"journal_article","file_date_updated":"2022-08-12T16:48:59Z","article_type":"original","user_id":"85279","department":[{"_id":"636"}],"_id":"29236","citation":{"ama":"McLachlan R, Offen C. Backward error analysis for conjugate symplectic methods. <i>Journal of Geometric Mechanics</i>. 2023;15(1):98-115. doi:<a href=\"https://doi.org/10.3934/jgm.2023005\">10.3934/jgm.2023005</a>","chicago":"McLachlan, Robert, and Christian Offen. “Backward Error Analysis for Conjugate Symplectic Methods.” <i>Journal of Geometric Mechanics</i> 15, no. 1 (2023): 98–115. <a href=\"https://doi.org/10.3934/jgm.2023005\">https://doi.org/10.3934/jgm.2023005</a>.","ieee":"R. McLachlan and C. Offen, “Backward error analysis for conjugate symplectic methods,” <i>Journal of Geometric Mechanics</i>, vol. 15, no. 1, pp. 98–115, 2023, doi: <a href=\"https://doi.org/10.3934/jgm.2023005\">10.3934/jgm.2023005</a>.","mla":"McLachlan, Robert, and Christian Offen. “Backward Error Analysis for Conjugate Symplectic Methods.” <i>Journal of Geometric Mechanics</i>, vol. 15, no. 1, AIMS Press, 2023, pp. 98–115, doi:<a href=\"https://doi.org/10.3934/jgm.2023005\">10.3934/jgm.2023005</a>.","short":"R. McLachlan, C. Offen, Journal of Geometric Mechanics 15 (2023) 98–115.","bibtex":"@article{McLachlan_Offen_2023, title={Backward error analysis for conjugate symplectic methods}, volume={15}, DOI={<a href=\"https://doi.org/10.3934/jgm.2023005\">10.3934/jgm.2023005</a>}, number={1}, journal={Journal of Geometric Mechanics}, publisher={AIMS Press}, author={McLachlan, Robert and Offen, Christian}, year={2023}, pages={98–115} }","apa":"McLachlan, R., &#38; Offen, C. (2023). Backward error analysis for conjugate symplectic methods. <i>Journal of Geometric Mechanics</i>, <i>15</i>(1), 98–115. <a href=\"https://doi.org/10.3934/jgm.2023005\">https://doi.org/10.3934/jgm.2023005</a>"},"intvolume":"        15","page":"98-115","related_material":{"link":[{"relation":"software","url":"https://github.com/Christian-Offen/BEAConjugateSymplectic"}]},"publication_status":"published","has_accepted_license":"1","doi":"10.3934/jgm.2023005","author":[{"first_name":"Robert","last_name":"McLachlan","full_name":"McLachlan, Robert"},{"orcid":"0000-0002-5940-8057","last_name":"Offen","full_name":"Offen, Christian","id":"85279","first_name":"Christian"}],"volume":15,"date_updated":"2023-08-10T08:40:30Z","oa":"1"},{"citation":{"chicago":"Bo, Fu, Li Ting, and Tobias Hemsel. “A Simple Pre-Stress Estimating Method of Langevin Transducers.” In <i>Piezoelectricity, Acoustic Waves, and Device Applications, 2008. SPAWDA 2008. Symposium On</i>, 324–27, 2008. <a href=\"https://doi.org/10.1109/SPAWDA.2008.4775801\">https://doi.org/10.1109/SPAWDA.2008.4775801</a>.","ieee":"F. Bo, L. Ting, and T. Hemsel, “A simple pre-stress estimating method of langevin transducers,” in <i>Piezoelectricity, Acoustic Waves, and Device Applications, 2008. SPAWDA 2008. Symposium on</i>, 2008, pp. 324–327.","ama":"Bo F, Ting L, Hemsel T. A simple pre-stress estimating method of langevin transducers. In: <i>Piezoelectricity, Acoustic Waves, and Device Applications, 2008. SPAWDA 2008. Symposium On</i>. ; 2008:324-327. doi:<a href=\"https://doi.org/10.1109/SPAWDA.2008.4775801\">10.1109/SPAWDA.2008.4775801</a>","mla":"Bo, Fu, et al. “A Simple Pre-Stress Estimating Method of Langevin Transducers.” <i>Piezoelectricity, Acoustic Waves, and Device Applications, 2008. SPAWDA 2008. Symposium On</i>, 2008, pp. 324–27, doi:<a href=\"https://doi.org/10.1109/SPAWDA.2008.4775801\">10.1109/SPAWDA.2008.4775801</a>.","bibtex":"@inproceedings{Bo_Ting_Hemsel_2008, title={A simple pre-stress estimating method of langevin transducers}, DOI={<a href=\"https://doi.org/10.1109/SPAWDA.2008.4775801\">10.1109/SPAWDA.2008.4775801</a>}, booktitle={Piezoelectricity, Acoustic Waves, and Device Applications, 2008. SPAWDA 2008. Symposium on}, author={Bo, Fu and Ting, Li and Hemsel, Tobias}, year={2008}, pages={324–327} }","short":"F. Bo, L. Ting, T. Hemsel, in: Piezoelectricity, Acoustic Waves, and Device Applications, 2008. SPAWDA 2008. Symposium On, 2008, pp. 324–327.","apa":"Bo, F., Ting, L., &#38; Hemsel, T. (2008). A simple pre-stress estimating method of langevin transducers. In <i>Piezoelectricity, Acoustic Waves, and Device Applications, 2008. SPAWDA 2008. Symposium on</i> (pp. 324–327). <a href=\"https://doi.org/10.1109/SPAWDA.2008.4775801\">https://doi.org/10.1109/SPAWDA.2008.4775801</a>"},"page":"324-327","year":"2008","author":[{"last_name":"Bo","full_name":"Bo, Fu","first_name":"Fu"},{"first_name":"Li","last_name":"Ting","full_name":"Ting, Li"},{"first_name":"Tobias","last_name":"Hemsel","id":"210","full_name":"Hemsel, Tobias"}],"date_created":"2019-04-29T11:16:13Z","date_updated":"2022-01-06T07:04:16Z","doi":"10.1109/SPAWDA.2008.4775801","title":"A simple pre-stress estimating method of langevin transducers","type":"conference","publication":"Piezoelectricity, Acoustic Waves, and Device Applications, 2008. SPAWDA 2008. Symposium on","status":"public","abstract":[{"text":"A simple pre-stress estimate method of Langevin transducers is studied. The measurement setup consists of a capacitor, an impedance converter and a voltmeter. Based on the piezoelectric equation and the basic circuit theory, the mathematical expression between the pre-stress and the voltage across the capacitor is derived. The pre-stress level can then be calculated out of the measurement of the capacitor voltage. In order to evaluate the precision of this method, a force washer is used to measure the pre-stress of the Langevin transducer. The result shows the pre-stress level obtained from this method is 30-40\\% higher than the pre-stress level measured by the force washer. This method is simple and can be used to estimate the pre-stress of various Langevin transducers. The precision of this method can be raised if d33 is identified under different pre-stress levels.","lang":"eng"}],"user_id":"55222","department":[{"_id":"151"}],"_id":"9568","language":[{"iso":"eng"}],"keyword":["capacitors","impedance convertors","piezoelectric transducers","stress analysis","Langevin transducers","basic circuit theory","capacitor","impedance converter","piezoelectric equation","pre-stress estimating method","voltmeter","Capacitors","Educational institutions","Equations","Force measurement","Impedance measurement","Manufacturing","Mechatronics","Piezoelectric transducers","Voltage","Voltmeters","Langevin transducer","capacitor","piezoelectric element","pre-stress"]}]