Learning on the Temporal Tangent Bundle for Physics-Informed Neural Networks
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866908960450150400 |
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| author | Jamal, Adetola Charbel, Mamlankou Wilfrid, Houédanou Koffi Guy, Dègla Aymard |
| author_facet | Jamal, Adetola Charbel, Mamlankou Wilfrid, Houédanou Koffi Guy, Dègla Aymard |
| contents | This paper addresses the limitations of Physics-Informed Neural Networks for time-dependent problems by introducing a tangent bundle learning framework. Instead of directly approximating the solution, we parameterize its temporal derivative and reconstruct the state through a Volterra integral operator that enforces initial conditions exactly. This approach eliminates competing soft constraints and naturally amplifies high-frequency errors through differentiation, countering spectral bias. We prove theoretical equivalence between minimizing the differentiated residual and solving the original partial differential equation. Experiments on advection, Burgers, and Klein-Gordon equations show that the proposed method achieves 100 to 200 times lower errors than standard approaches using compact three-layer networks, with superior shock-capturing and long-time accuracy. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_11829 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Learning on the Temporal Tangent Bundle for Physics-Informed Neural Networks Jamal, Adetola Charbel, Mamlankou Wilfrid, Houédanou Koffi Guy, Dègla Aymard Numerical Analysis 65M99, 35K05, 35Q53, 68T07 This paper addresses the limitations of Physics-Informed Neural Networks for time-dependent problems by introducing a tangent bundle learning framework. Instead of directly approximating the solution, we parameterize its temporal derivative and reconstruct the state through a Volterra integral operator that enforces initial conditions exactly. This approach eliminates competing soft constraints and naturally amplifies high-frequency errors through differentiation, countering spectral bias. We prove theoretical equivalence between minimizing the differentiated residual and solving the original partial differential equation. Experiments on advection, Burgers, and Klein-Gordon equations show that the proposed method achieves 100 to 200 times lower errors than standard approaches using compact three-layer networks, with superior shock-capturing and long-time accuracy. |
| title | Learning on the Temporal Tangent Bundle for Physics-Informed Neural Networks |
| topic | Numerical Analysis 65M99, 35K05, 35Q53, 68T07 |
| url | https://arxiv.org/abs/2604.11829 |