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Journal : progressive physics journal

Application of Physics-Informed Neural Networks (PINNs) for the Numerical Solution of the Time-Independent Schrödinger Equation Akhsan, Hamdi; Khoirun Nisa; Nurhikmah, Putri; Wailaina; Ariska, Melly
Progressive Physics Journal Vol. 6 No. 2 (2025): Progressive Physics Journal
Publisher : Program Studi Fisika, Jurusan Fisika, FMIPA, Universitas Mulawarman

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.30872/6881zx10

Abstract

This work investigates the application of Physics-Informed Neural Networks (PINNs) for numerical solutions to the time-independent Schrödinger equation of the quantum harmonic oscillator in one, two, and three spatial dimensions. Fully connected neural architectures are constructed to approximate wavefunctions over finite symmetric domains, while the corresponding energy eigenvalues are treated as trainable parameters. The training strategy utilizes randomly sampled interior points to enforce the Schrödinger operator residual and boundary points to impose vanishing wavefunction constraints. For the 1D quantum harmonic oscillator, the learned ground-state wavefunction yields an energy of E = 1.2939 after 12,000 iterations. In the 2D configuration, convergence is achieved at E = 2.1352 within 14,000 iterations, whereas the 3D model attains E = 2.6377 after 12,000 iterations. These values agree with the expected trend of increasing ground-state energy with dimensionality, although deviations from exact analytical values indicate that PINNs may experience optimization challenges and sensitivity to sampling density and boundary enforcement. Despite these limitations, the trained models successfully capture the characteristic spatial symmetries and Gaussian-like envelope of harmonic oscillator eigenstates across all dimensions. These findings demonstrate that PINNs offer a flexible, mesh-free alternative for solving stationary quantum systems, particularly when analytical or conventional numerical approaches become impractical. The method shows strong potential for higher-dimensional quantum applications, even though further refinement such as improved sampling, loss balancing, and network depth remains necessary to suppress residual error and enhance eigenvalue accuracy.
Co-Authors Abidin Pasaribu Abidin Pasaribu Abidin Pasaribu, Abidin Adam Darmawan Ade Kurniawan Ade Kurniawan Agustina, Atika Ahmad Firdaus Falka Ahmad Fitra Ritonga Aisyah Veronika Al Fatih, Zaky Alawiyah, Sakinah Alfin Suhanda Amalia, Wardah Amanda Rahmannisa Amanda, Karenina Andriani, Nelly Apit Fathurohman Ari Widodo Arini Rosa Sinensis Atika Agustina Az Zahra, Lutfiah Azizah Putri Berimah Azizah Putri Berimah Berimah, Azizah Putri Dea Amalia Dea Rizky Oktavia Diah Kartika Sari Dina Maulina Dwi Purnomo Aji Dwicahyani, Rania Dwita Kartika Sari Edo Anugrah Efrinalia, Winta Ernalida Ernalida Evelina Astra Patriot Fena Siska Putriyani Firdausi, Huriyatul Fitra Ritonga, Ahmad Fitriyani Fitriyani Frendi Ihwan Syamsudin Frida Ramadian Gelby Pradina Paramitha Gisca Rivia Prameswari Hamdi Akhsan Hamdi Akhsan Hartono Hartono Hartono Hartono Hartono Hartono Hartono Hartono Hepiyani Hepiyani Herlambang, Dominikus Krisna Ida Sriyanti Ida Sriyanti Iful Amri Iin Seprina Iskhaq Iskandar Ismet Ismet Ismet, Ismet Jesi Pebralia Ketang Wiyono Ketang Wiyono Ketang Wiyono Ketang Wiyono KHOIRUN NISA Kistiono Kistiono Kistiono Kistiono Kristylia Sury Laras Sapitri, Cindy Laura Insan Kencana Leni Marlina Lira Diska Wati Manurung, Nia Three Markos Siahaan, Sardianto Masduki Masduki May Sari Mega Fatimah Rosana Melati, Pegi Meli Asma Desti Melvany, Nanda Eva Milka, Ikbal Adrian Mindia Vanessa Pratiwi, Sri Muhamad Yusup Muhamad Yusup Muhammad afrizal Muhammad Afrizal Muhammad Aufa Riyaldo Muhammad Irfan Muhammad Irfan muhammad muslim Muhammad Muslim Muhammad Muslim Muhammad Muslim, Muhammad Muhammad Putra Aririyansyah Muhammad Romadoni Muhammad Yusuf Muhammad Yusup Mulyadi Eko Purnomo, Mulyadi Eko Murnia Murniati Murniati . Muslimah, Resta Ulis Nadiar Pertiwi Nely Andriani Nely Andriani, Nely Nilam Cahyati Nor Farahwahidah Novi Yusliani Nur Julia Ningsih Nur Kayla Lekati Nurhikmah, Putri Nurjannah Nurjannah Nuzula, Khalidatun Prajaska, Nicola Pratiwi Ineke Anwar Putra, Guruh Sukarno Putri Maulida, Nabila Putri, Astrid Yulinda Putri, Jamiatul Khairunnisa Putriyani, Fena Siska Rahmannisa, Amanda Rahmi Susanti Rahmi Susanti Rahmi Susanti Rahmi, Ani Ramadhani, Neysya Ditha Rara Rara Ratu Ilma Indra Putri Redondo, Fernando Eric Rini Khoirunnisa Rita Inderawati Ritonga, Ahmad Fitra Rizki Novianti, Rizki Romadoni, Muhammad Ruth Magdalena Suryaningsih Sakinah Alawiyah Salmah Rianti Salsabillah, Zahwa Nur Saparini Saparini Sardianto Markos Siahaan Sardianto Markos Siahaan, Sardianto Markos Sari, Dwita Kartika Sary Silvhiany Sayyendra, Amelia Putri Senandung Senja Citra Maharani Seprina, Iin Setiyowati, Devi Ariska Siahaan, Sardianto Marcos Siregar, Anpo Ferdy Siti Nur Azizah Sopia Sopia Sri Mindia Vanessa Pratiwi Sri Zakiyah Sudirman Sudirman Sudirman Sudirman Sudirman, Sudirman Suhadi Suhadi Suhanda, Alfin Sunyono - - Supari Supari Supari Supari, Supari Suryaningsih, Ruth Magdalena Syahrani, Revanola Gusti Syarifudin, Agus Syuhendri, Syuhendri Taufiq Taufiq Tine Aprianti Tita Ratna Wulan Dari Tsabita Husna Tyse Ramadhona Utami, Amanda Kurnia Viyanti Viyanti Wailaina Wati, Lira Diska Yasmine Altira Yenny Anwar Zahlia, Marsela Zahra Alwi, Zahra Zahra Ayanna Firmansyah Zulherman Zulherman Zulherman Zulherman, Zulherman