A novel sequential method to train physics informed neural networks for Allen Cahn and Cahn Hilliard equations
A physics informed neural network (PINN) incorporates the physics of a system by satisfying its boundary value problem through a neural network’s loss function. The PINN approach has shown great success in approximating the map between the solution of a partial differential equation (PDE) and its sp...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Mattey, Revanth [verfasserIn] |
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| Format: |
E-Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2022transfer abstract |
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| Schlagwörter: |
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| Übergeordnetes Werk: |
Enthalten in: Does enhanced hydration have impact on autogenous deformation of internally cued mortar? - Zou, Dinghua ELSEVIER, 2019, Amsterdam [u.a.] |
|---|---|
| Übergeordnetes Werk: |
volume:390 ; year:2022 ; day:15 ; month:02 ; pages:0 |
| Links: |
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| DOI / URN: |
10.1016/j.cma.2021.114474 |
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ELV056562160 |
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| 520 | |a A physics informed neural network (PINN) incorporates the physics of a system by satisfying its boundary value problem through a neural network’s loss function. The PINN approach has shown great success in approximating the map between the solution of a partial differential equation (PDE) and its spatio-temporal coordinates. However, the PINN’s accuracy suffers significantly for strongly non-linear and higher-order time-varying partial differential equations such as Allen Cahn and Cahn Hilliard equations. To resolve this problem, a novel PINN scheme is proposed that solves the PDE sequentially over successive time segments using a single neural network. The key idea is to re-train the same neural network for solving the PDE over successive time segments while satisfying the already obtained solution for all previous time segments. Thus it is named as backward compatible PINN (bc-PINN). To illustrate the advantages of bc-PINN, the Cahn Hilliard and Allen Cahn equations are solved. These equations are widely used to describe phase separation and reaction–diffusion systems. Additionally, two new techniques have been introduced to improve the proposed bc-PINN scheme. The first technique uses the initial condition of a time-segment to guide the neural network map closer to the true map over that segment. The second technique is a transfer learning approach where the features learned from the previous training are preserved. We have demonstrated that these two techniques improve the accuracy and efficiency of the bc-PINN scheme significantly. It has also been demonstrated that the convergence is improved by using a phase space representation for higher-order PDEs. It is shown that the proposed bc-PINN technique is significantly more accurate and efficient than PINN. | ||
| 520 | |a A physics informed neural network (PINN) incorporates the physics of a system by satisfying its boundary value problem through a neural network’s loss function. The PINN approach has shown great success in approximating the map between the solution of a partial differential equation (PDE) and its spatio-temporal coordinates. However, the PINN’s accuracy suffers significantly for strongly non-linear and higher-order time-varying partial differential equations such as Allen Cahn and Cahn Hilliard equations. To resolve this problem, a novel PINN scheme is proposed that solves the PDE sequentially over successive time segments using a single neural network. The key idea is to re-train the same neural network for solving the PDE over successive time segments while satisfying the already obtained solution for all previous time segments. Thus it is named as backward compatible PINN (bc-PINN). To illustrate the advantages of bc-PINN, the Cahn Hilliard and Allen Cahn equations are solved. These equations are widely used to describe phase separation and reaction–diffusion systems. Additionally, two new techniques have been introduced to improve the proposed bc-PINN scheme. The first technique uses the initial condition of a time-segment to guide the neural network map closer to the true map over that segment. The second technique is a transfer learning approach where the features learned from the previous training are preserved. We have demonstrated that these two techniques improve the accuracy and efficiency of the bc-PINN scheme significantly. It has also been demonstrated that the convergence is improved by using a phase space representation for higher-order PDEs. It is shown that the proposed bc-PINN technique is significantly more accurate and efficient than PINN. | ||
| 650 | 7 | |a Physics informed neural networks |2 Elsevier | |
| 650 | 7 | |a Cahn Hilliard equation |2 Elsevier | |
| 650 | 7 | |a Allen Cahn equation |2 Elsevier | |
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10.1016/j.cma.2021.114474 doi /cbs_pica/cbs_olc/import_discovery/elsevier/einzuspielen/GBV00000000001650.pica (DE-627)ELV056562160 (ELSEVIER)S0045-7825(21)00693-9 DE-627 ger DE-627 rakwb eng 690 VZ 56.45 bkl Mattey, Revanth verfasserin aut A novel sequential method to train physics informed neural networks for Allen Cahn and Cahn Hilliard equations 2022transfer abstract nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier A physics informed neural network (PINN) incorporates the physics of a system by satisfying its boundary value problem through a neural network’s loss function. The PINN approach has shown great success in approximating the map between the solution of a partial differential equation (PDE) and its spatio-temporal coordinates. However, the PINN’s accuracy suffers significantly for strongly non-linear and higher-order time-varying partial differential equations such as Allen Cahn and Cahn Hilliard equations. To resolve this problem, a novel PINN scheme is proposed that solves the PDE sequentially over successive time segments using a single neural network. The key idea is to re-train the same neural network for solving the PDE over successive time segments while satisfying the already obtained solution for all previous time segments. Thus it is named as backward compatible PINN (bc-PINN). To illustrate the advantages of bc-PINN, the Cahn Hilliard and Allen Cahn equations are solved. These equations are widely used to describe phase separation and reaction–diffusion systems. Additionally, two new techniques have been introduced to improve the proposed bc-PINN scheme. The first technique uses the initial condition of a time-segment to guide the neural network map closer to the true map over that segment. The second technique is a transfer learning approach where the features learned from the previous training are preserved. We have demonstrated that these two techniques improve the accuracy and efficiency of the bc-PINN scheme significantly. It has also been demonstrated that the convergence is improved by using a phase space representation for higher-order PDEs. It is shown that the proposed bc-PINN technique is significantly more accurate and efficient than PINN. A physics informed neural network (PINN) incorporates the physics of a system by satisfying its boundary value problem through a neural network’s loss function. The PINN approach has shown great success in approximating the map between the solution of a partial differential equation (PDE) and its spatio-temporal coordinates. However, the PINN’s accuracy suffers significantly for strongly non-linear and higher-order time-varying partial differential equations such as Allen Cahn and Cahn Hilliard equations. To resolve this problem, a novel PINN scheme is proposed that solves the PDE sequentially over successive time segments using a single neural network. The key idea is to re-train the same neural network for solving the PDE over successive time segments while satisfying the already obtained solution for all previous time segments. Thus it is named as backward compatible PINN (bc-PINN). To illustrate the advantages of bc-PINN, the Cahn Hilliard and Allen Cahn equations are solved. These equations are widely used to describe phase separation and reaction–diffusion systems. Additionally, two new techniques have been introduced to improve the proposed bc-PINN scheme. The first technique uses the initial condition of a time-segment to guide the neural network map closer to the true map over that segment. The second technique is a transfer learning approach where the features learned from the previous training are preserved. We have demonstrated that these two techniques improve the accuracy and efficiency of the bc-PINN scheme significantly. It has also been demonstrated that the convergence is improved by using a phase space representation for higher-order PDEs. It is shown that the proposed bc-PINN technique is significantly more accurate and efficient than PINN. Physics informed neural networks Elsevier Cahn Hilliard equation Elsevier Allen Cahn equation Elsevier Partial differential equation (PDEs) Elsevier Ghosh, Susanta oth Enthalten in Elsevier Science Zou, Dinghua ELSEVIER Does enhanced hydration have impact on autogenous deformation of internally cued mortar? 2019 Amsterdam [u.a.] (DE-627)ELV002113945 volume:390 year:2022 day:15 month:02 pages:0 https://doi.org/10.1016/j.cma.2021.114474 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U 56.45 Baustoffkunde VZ AR 390 2022 15 0215 0 |
| spelling |
10.1016/j.cma.2021.114474 doi /cbs_pica/cbs_olc/import_discovery/elsevier/einzuspielen/GBV00000000001650.pica (DE-627)ELV056562160 (ELSEVIER)S0045-7825(21)00693-9 DE-627 ger DE-627 rakwb eng 690 VZ 56.45 bkl Mattey, Revanth verfasserin aut A novel sequential method to train physics informed neural networks for Allen Cahn and Cahn Hilliard equations 2022transfer abstract nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier A physics informed neural network (PINN) incorporates the physics of a system by satisfying its boundary value problem through a neural network’s loss function. The PINN approach has shown great success in approximating the map between the solution of a partial differential equation (PDE) and its spatio-temporal coordinates. However, the PINN’s accuracy suffers significantly for strongly non-linear and higher-order time-varying partial differential equations such as Allen Cahn and Cahn Hilliard equations. To resolve this problem, a novel PINN scheme is proposed that solves the PDE sequentially over successive time segments using a single neural network. The key idea is to re-train the same neural network for solving the PDE over successive time segments while satisfying the already obtained solution for all previous time segments. Thus it is named as backward compatible PINN (bc-PINN). To illustrate the advantages of bc-PINN, the Cahn Hilliard and Allen Cahn equations are solved. These equations are widely used to describe phase separation and reaction–diffusion systems. Additionally, two new techniques have been introduced to improve the proposed bc-PINN scheme. The first technique uses the initial condition of a time-segment to guide the neural network map closer to the true map over that segment. The second technique is a transfer learning approach where the features learned from the previous training are preserved. We have demonstrated that these two techniques improve the accuracy and efficiency of the bc-PINN scheme significantly. It has also been demonstrated that the convergence is improved by using a phase space representation for higher-order PDEs. It is shown that the proposed bc-PINN technique is significantly more accurate and efficient than PINN. A physics informed neural network (PINN) incorporates the physics of a system by satisfying its boundary value problem through a neural network’s loss function. The PINN approach has shown great success in approximating the map between the solution of a partial differential equation (PDE) and its spatio-temporal coordinates. However, the PINN’s accuracy suffers significantly for strongly non-linear and higher-order time-varying partial differential equations such as Allen Cahn and Cahn Hilliard equations. To resolve this problem, a novel PINN scheme is proposed that solves the PDE sequentially over successive time segments using a single neural network. The key idea is to re-train the same neural network for solving the PDE over successive time segments while satisfying the already obtained solution for all previous time segments. Thus it is named as backward compatible PINN (bc-PINN). To illustrate the advantages of bc-PINN, the Cahn Hilliard and Allen Cahn equations are solved. These equations are widely used to describe phase separation and reaction–diffusion systems. Additionally, two new techniques have been introduced to improve the proposed bc-PINN scheme. The first technique uses the initial condition of a time-segment to guide the neural network map closer to the true map over that segment. The second technique is a transfer learning approach where the features learned from the previous training are preserved. We have demonstrated that these two techniques improve the accuracy and efficiency of the bc-PINN scheme significantly. It has also been demonstrated that the convergence is improved by using a phase space representation for higher-order PDEs. It is shown that the proposed bc-PINN technique is significantly more accurate and efficient than PINN. Physics informed neural networks Elsevier Cahn Hilliard equation Elsevier Allen Cahn equation Elsevier Partial differential equation (PDEs) Elsevier Ghosh, Susanta oth Enthalten in Elsevier Science Zou, Dinghua ELSEVIER Does enhanced hydration have impact on autogenous deformation of internally cued mortar? 2019 Amsterdam [u.a.] (DE-627)ELV002113945 volume:390 year:2022 day:15 month:02 pages:0 https://doi.org/10.1016/j.cma.2021.114474 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U 56.45 Baustoffkunde VZ AR 390 2022 15 0215 0 |
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10.1016/j.cma.2021.114474 doi /cbs_pica/cbs_olc/import_discovery/elsevier/einzuspielen/GBV00000000001650.pica (DE-627)ELV056562160 (ELSEVIER)S0045-7825(21)00693-9 DE-627 ger DE-627 rakwb eng 690 VZ 56.45 bkl Mattey, Revanth verfasserin aut A novel sequential method to train physics informed neural networks for Allen Cahn and Cahn Hilliard equations 2022transfer abstract nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier A physics informed neural network (PINN) incorporates the physics of a system by satisfying its boundary value problem through a neural network’s loss function. The PINN approach has shown great success in approximating the map between the solution of a partial differential equation (PDE) and its spatio-temporal coordinates. However, the PINN’s accuracy suffers significantly for strongly non-linear and higher-order time-varying partial differential equations such as Allen Cahn and Cahn Hilliard equations. To resolve this problem, a novel PINN scheme is proposed that solves the PDE sequentially over successive time segments using a single neural network. The key idea is to re-train the same neural network for solving the PDE over successive time segments while satisfying the already obtained solution for all previous time segments. Thus it is named as backward compatible PINN (bc-PINN). To illustrate the advantages of bc-PINN, the Cahn Hilliard and Allen Cahn equations are solved. These equations are widely used to describe phase separation and reaction–diffusion systems. Additionally, two new techniques have been introduced to improve the proposed bc-PINN scheme. The first technique uses the initial condition of a time-segment to guide the neural network map closer to the true map over that segment. The second technique is a transfer learning approach where the features learned from the previous training are preserved. We have demonstrated that these two techniques improve the accuracy and efficiency of the bc-PINN scheme significantly. It has also been demonstrated that the convergence is improved by using a phase space representation for higher-order PDEs. It is shown that the proposed bc-PINN technique is significantly more accurate and efficient than PINN. A physics informed neural network (PINN) incorporates the physics of a system by satisfying its boundary value problem through a neural network’s loss function. The PINN approach has shown great success in approximating the map between the solution of a partial differential equation (PDE) and its spatio-temporal coordinates. However, the PINN’s accuracy suffers significantly for strongly non-linear and higher-order time-varying partial differential equations such as Allen Cahn and Cahn Hilliard equations. To resolve this problem, a novel PINN scheme is proposed that solves the PDE sequentially over successive time segments using a single neural network. The key idea is to re-train the same neural network for solving the PDE over successive time segments while satisfying the already obtained solution for all previous time segments. Thus it is named as backward compatible PINN (bc-PINN). To illustrate the advantages of bc-PINN, the Cahn Hilliard and Allen Cahn equations are solved. These equations are widely used to describe phase separation and reaction–diffusion systems. Additionally, two new techniques have been introduced to improve the proposed bc-PINN scheme. The first technique uses the initial condition of a time-segment to guide the neural network map closer to the true map over that segment. The second technique is a transfer learning approach where the features learned from the previous training are preserved. We have demonstrated that these two techniques improve the accuracy and efficiency of the bc-PINN scheme significantly. It has also been demonstrated that the convergence is improved by using a phase space representation for higher-order PDEs. It is shown that the proposed bc-PINN technique is significantly more accurate and efficient than PINN. Physics informed neural networks Elsevier Cahn Hilliard equation Elsevier Allen Cahn equation Elsevier Partial differential equation (PDEs) Elsevier Ghosh, Susanta oth Enthalten in Elsevier Science Zou, Dinghua ELSEVIER Does enhanced hydration have impact on autogenous deformation of internally cued mortar? 2019 Amsterdam [u.a.] (DE-627)ELV002113945 volume:390 year:2022 day:15 month:02 pages:0 https://doi.org/10.1016/j.cma.2021.114474 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U 56.45 Baustoffkunde VZ AR 390 2022 15 0215 0 |
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10.1016/j.cma.2021.114474 doi /cbs_pica/cbs_olc/import_discovery/elsevier/einzuspielen/GBV00000000001650.pica (DE-627)ELV056562160 (ELSEVIER)S0045-7825(21)00693-9 DE-627 ger DE-627 rakwb eng 690 VZ 56.45 bkl Mattey, Revanth verfasserin aut A novel sequential method to train physics informed neural networks for Allen Cahn and Cahn Hilliard equations 2022transfer abstract nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier A physics informed neural network (PINN) incorporates the physics of a system by satisfying its boundary value problem through a neural network’s loss function. The PINN approach has shown great success in approximating the map between the solution of a partial differential equation (PDE) and its spatio-temporal coordinates. However, the PINN’s accuracy suffers significantly for strongly non-linear and higher-order time-varying partial differential equations such as Allen Cahn and Cahn Hilliard equations. To resolve this problem, a novel PINN scheme is proposed that solves the PDE sequentially over successive time segments using a single neural network. The key idea is to re-train the same neural network for solving the PDE over successive time segments while satisfying the already obtained solution for all previous time segments. Thus it is named as backward compatible PINN (bc-PINN). To illustrate the advantages of bc-PINN, the Cahn Hilliard and Allen Cahn equations are solved. These equations are widely used to describe phase separation and reaction–diffusion systems. Additionally, two new techniques have been introduced to improve the proposed bc-PINN scheme. The first technique uses the initial condition of a time-segment to guide the neural network map closer to the true map over that segment. The second technique is a transfer learning approach where the features learned from the previous training are preserved. We have demonstrated that these two techniques improve the accuracy and efficiency of the bc-PINN scheme significantly. It has also been demonstrated that the convergence is improved by using a phase space representation for higher-order PDEs. It is shown that the proposed bc-PINN technique is significantly more accurate and efficient than PINN. A physics informed neural network (PINN) incorporates the physics of a system by satisfying its boundary value problem through a neural network’s loss function. The PINN approach has shown great success in approximating the map between the solution of a partial differential equation (PDE) and its spatio-temporal coordinates. However, the PINN’s accuracy suffers significantly for strongly non-linear and higher-order time-varying partial differential equations such as Allen Cahn and Cahn Hilliard equations. To resolve this problem, a novel PINN scheme is proposed that solves the PDE sequentially over successive time segments using a single neural network. The key idea is to re-train the same neural network for solving the PDE over successive time segments while satisfying the already obtained solution for all previous time segments. Thus it is named as backward compatible PINN (bc-PINN). To illustrate the advantages of bc-PINN, the Cahn Hilliard and Allen Cahn equations are solved. These equations are widely used to describe phase separation and reaction–diffusion systems. Additionally, two new techniques have been introduced to improve the proposed bc-PINN scheme. The first technique uses the initial condition of a time-segment to guide the neural network map closer to the true map over that segment. The second technique is a transfer learning approach where the features learned from the previous training are preserved. We have demonstrated that these two techniques improve the accuracy and efficiency of the bc-PINN scheme significantly. It has also been demonstrated that the convergence is improved by using a phase space representation for higher-order PDEs. It is shown that the proposed bc-PINN technique is significantly more accurate and efficient than PINN. Physics informed neural networks Elsevier Cahn Hilliard equation Elsevier Allen Cahn equation Elsevier Partial differential equation (PDEs) Elsevier Ghosh, Susanta oth Enthalten in Elsevier Science Zou, Dinghua ELSEVIER Does enhanced hydration have impact on autogenous deformation of internally cued mortar? 2019 Amsterdam [u.a.] (DE-627)ELV002113945 volume:390 year:2022 day:15 month:02 pages:0 https://doi.org/10.1016/j.cma.2021.114474 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U 56.45 Baustoffkunde VZ AR 390 2022 15 0215 0 |
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10.1016/j.cma.2021.114474 doi /cbs_pica/cbs_olc/import_discovery/elsevier/einzuspielen/GBV00000000001650.pica (DE-627)ELV056562160 (ELSEVIER)S0045-7825(21)00693-9 DE-627 ger DE-627 rakwb eng 690 VZ 56.45 bkl Mattey, Revanth verfasserin aut A novel sequential method to train physics informed neural networks for Allen Cahn and Cahn Hilliard equations 2022transfer abstract nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier A physics informed neural network (PINN) incorporates the physics of a system by satisfying its boundary value problem through a neural network’s loss function. The PINN approach has shown great success in approximating the map between the solution of a partial differential equation (PDE) and its spatio-temporal coordinates. However, the PINN’s accuracy suffers significantly for strongly non-linear and higher-order time-varying partial differential equations such as Allen Cahn and Cahn Hilliard equations. To resolve this problem, a novel PINN scheme is proposed that solves the PDE sequentially over successive time segments using a single neural network. The key idea is to re-train the same neural network for solving the PDE over successive time segments while satisfying the already obtained solution for all previous time segments. Thus it is named as backward compatible PINN (bc-PINN). To illustrate the advantages of bc-PINN, the Cahn Hilliard and Allen Cahn equations are solved. These equations are widely used to describe phase separation and reaction–diffusion systems. Additionally, two new techniques have been introduced to improve the proposed bc-PINN scheme. The first technique uses the initial condition of a time-segment to guide the neural network map closer to the true map over that segment. The second technique is a transfer learning approach where the features learned from the previous training are preserved. We have demonstrated that these two techniques improve the accuracy and efficiency of the bc-PINN scheme significantly. It has also been demonstrated that the convergence is improved by using a phase space representation for higher-order PDEs. It is shown that the proposed bc-PINN technique is significantly more accurate and efficient than PINN. A physics informed neural network (PINN) incorporates the physics of a system by satisfying its boundary value problem through a neural network’s loss function. The PINN approach has shown great success in approximating the map between the solution of a partial differential equation (PDE) and its spatio-temporal coordinates. However, the PINN’s accuracy suffers significantly for strongly non-linear and higher-order time-varying partial differential equations such as Allen Cahn and Cahn Hilliard equations. To resolve this problem, a novel PINN scheme is proposed that solves the PDE sequentially over successive time segments using a single neural network. The key idea is to re-train the same neural network for solving the PDE over successive time segments while satisfying the already obtained solution for all previous time segments. Thus it is named as backward compatible PINN (bc-PINN). To illustrate the advantages of bc-PINN, the Cahn Hilliard and Allen Cahn equations are solved. These equations are widely used to describe phase separation and reaction–diffusion systems. Additionally, two new techniques have been introduced to improve the proposed bc-PINN scheme. The first technique uses the initial condition of a time-segment to guide the neural network map closer to the true map over that segment. The second technique is a transfer learning approach where the features learned from the previous training are preserved. We have demonstrated that these two techniques improve the accuracy and efficiency of the bc-PINN scheme significantly. It has also been demonstrated that the convergence is improved by using a phase space representation for higher-order PDEs. It is shown that the proposed bc-PINN technique is significantly more accurate and efficient than PINN. Physics informed neural networks Elsevier Cahn Hilliard equation Elsevier Allen Cahn equation Elsevier Partial differential equation (PDEs) Elsevier Ghosh, Susanta oth Enthalten in Elsevier Science Zou, Dinghua ELSEVIER Does enhanced hydration have impact on autogenous deformation of internally cued mortar? 2019 Amsterdam [u.a.] (DE-627)ELV002113945 volume:390 year:2022 day:15 month:02 pages:0 https://doi.org/10.1016/j.cma.2021.114474 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U 56.45 Baustoffkunde VZ AR 390 2022 15 0215 0 |
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A novel sequential method to train physics informed neural networks for Allen Cahn and Cahn Hilliard equations |
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A physics informed neural network (PINN) incorporates the physics of a system by satisfying its boundary value problem through a neural network’s loss function. The PINN approach has shown great success in approximating the map between the solution of a partial differential equation (PDE) and its spatio-temporal coordinates. However, the PINN’s accuracy suffers significantly for strongly non-linear and higher-order time-varying partial differential equations such as Allen Cahn and Cahn Hilliard equations. To resolve this problem, a novel PINN scheme is proposed that solves the PDE sequentially over successive time segments using a single neural network. The key idea is to re-train the same neural network for solving the PDE over successive time segments while satisfying the already obtained solution for all previous time segments. Thus it is named as backward compatible PINN (bc-PINN). To illustrate the advantages of bc-PINN, the Cahn Hilliard and Allen Cahn equations are solved. These equations are widely used to describe phase separation and reaction–diffusion systems. Additionally, two new techniques have been introduced to improve the proposed bc-PINN scheme. The first technique uses the initial condition of a time-segment to guide the neural network map closer to the true map over that segment. The second technique is a transfer learning approach where the features learned from the previous training are preserved. We have demonstrated that these two techniques improve the accuracy and efficiency of the bc-PINN scheme significantly. It has also been demonstrated that the convergence is improved by using a phase space representation for higher-order PDEs. It is shown that the proposed bc-PINN technique is significantly more accurate and efficient than PINN. |
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A physics informed neural network (PINN) incorporates the physics of a system by satisfying its boundary value problem through a neural network’s loss function. The PINN approach has shown great success in approximating the map between the solution of a partial differential equation (PDE) and its spatio-temporal coordinates. However, the PINN’s accuracy suffers significantly for strongly non-linear and higher-order time-varying partial differential equations such as Allen Cahn and Cahn Hilliard equations. To resolve this problem, a novel PINN scheme is proposed that solves the PDE sequentially over successive time segments using a single neural network. The key idea is to re-train the same neural network for solving the PDE over successive time segments while satisfying the already obtained solution for all previous time segments. Thus it is named as backward compatible PINN (bc-PINN). To illustrate the advantages of bc-PINN, the Cahn Hilliard and Allen Cahn equations are solved. These equations are widely used to describe phase separation and reaction–diffusion systems. Additionally, two new techniques have been introduced to improve the proposed bc-PINN scheme. The first technique uses the initial condition of a time-segment to guide the neural network map closer to the true map over that segment. The second technique is a transfer learning approach where the features learned from the previous training are preserved. We have demonstrated that these two techniques improve the accuracy and efficiency of the bc-PINN scheme significantly. It has also been demonstrated that the convergence is improved by using a phase space representation for higher-order PDEs. It is shown that the proposed bc-PINN technique is significantly more accurate and efficient than PINN. |
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A physics informed neural network (PINN) incorporates the physics of a system by satisfying its boundary value problem through a neural network’s loss function. The PINN approach has shown great success in approximating the map between the solution of a partial differential equation (PDE) and its spatio-temporal coordinates. However, the PINN’s accuracy suffers significantly for strongly non-linear and higher-order time-varying partial differential equations such as Allen Cahn and Cahn Hilliard equations. To resolve this problem, a novel PINN scheme is proposed that solves the PDE sequentially over successive time segments using a single neural network. The key idea is to re-train the same neural network for solving the PDE over successive time segments while satisfying the already obtained solution for all previous time segments. Thus it is named as backward compatible PINN (bc-PINN). To illustrate the advantages of bc-PINN, the Cahn Hilliard and Allen Cahn equations are solved. These equations are widely used to describe phase separation and reaction–diffusion systems. Additionally, two new techniques have been introduced to improve the proposed bc-PINN scheme. The first technique uses the initial condition of a time-segment to guide the neural network map closer to the true map over that segment. The second technique is a transfer learning approach where the features learned from the previous training are preserved. We have demonstrated that these two techniques improve the accuracy and efficiency of the bc-PINN scheme significantly. It has also been demonstrated that the convergence is improved by using a phase space representation for higher-order PDEs. It is shown that the proposed bc-PINN technique is significantly more accurate and efficient than PINN. |
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