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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 Agung Mataram Agustina, Atika Ahmad Fauzi Ismail Ahmad Fudholi Aisyah Veronika Aji, Dwi Purnomo Aljuzza, Muhammad Ikhsan Almafie, Muhammad Rama Amalia, Wardah Amanda Kurnia Utami Amanda Rahmannisa Andriani, Nelly Anggelina, Putri Anggelina, Zanne Anisa Qoriq Atun Nasiroh Apit Fathurohman Apit Fathurohman Arini Rosa Sinensis Atika Agustina Aulia, Maghfira Ayu Sekar Sari Azizah Putri Berimah Berimah, Azizah Putri Cahyani, Delvina Putri Cahyati, Nilam Damayanti, Fitria Siska Dani, Rahma Dea Amalia Dea Rizky Oktavia Deni Sadly Detrika Mawar Arrum Devi Ardiantini Dia Oktarina Gumanti Dwi Purnomo Aji Dwi Purnomo Aji Dwicahyani, Rania Dwita Kartika Sari Edo Anugrah Edray, Auliya Einsrizkianne Elsie Sulastry Erazando Alfa Seira Ermalia, Widia Fahreza, Muhammad Aditya Fajriah, Nurkholisa Fatmawati, Fatmawati Fatonah, Amelia Febrianti, Tanti Fena Siska Putriyani Fika Nurul Hidayati Fina Maliya Firdausi, Huriyatul Fitria Siska damayanti Fitriyani Fitriyani Guruh Sukarno Putra Halim, Noor Dayana Abd Hayati, Suci Husnul I. Ismet Ida Sriyanti Iful Amri Iin Seprina Ina Agustina Intan Mahdiah Intan Purnama Sari Irenta Irenta Ishkaq Iskandar Iskhaq Iskandar Iskhaq Iskandar Iskhaq Iskandar Iskhaq Iskandar Ismet Ismet Ismet Ismet, Ismet Ismet, I. Izzah Nurhidayah, Izzah Jamiatul Khairunnisa Putri Jesi Pebralia Juliansyah, Dendi Juwita Febrianti Kartika, Nadia Dwi Ketang Wiyono Ketang Wiyono Ketang Wiyono KHOIRUN NISA Kistiono Kistiono Kistiono Kistiono Kistiono Kistiono Kistiono Kistiono Kistiono Komarudin Komarudin Kurnia, Mega Lamura, M. Danny Pratama Lara Sania Laras Sapitri, Cindy Leni Marlina Leni Marlina Leni Marlina Lestari, Mutiah Lestari, Yora Inda Lira Diska Wati Lira Lira Lira, Lira Lismayana, Pegi Putri M. Nicola Prajaska MAHARANI, MELINDA Mahdiah, Intan Mahmudah, Indri Maliya, Fina Manurung, Nia Three Markos Siahaan, Sardianto Mega Kurnia Mega Kurnia Meilanie, Trista Melly Ariska Melvany, Nanda Eva Miftahul Jannah Mindia Vanessa Pratiwi, Sri Muhamad Yusup Muhammad Aditiyah Muhammad Afrizal Muhammad afrizal Muhammad Aufa Riyaldo Muhammad Irfan Muhammad Irfan Muhammad Irfan Muhammad Irfan Muhammad Irfan muhammad muslim Muhammad Muslim Muhammad Muslim Muhammad Muslim, Muhammad Muhammad Romadoni Muhammad Romadoni Muhammad Yusuf Muhammad Yusup Murnia Murniati Murniati . Muslimah, Resta Ulis Mutia Faradini Ramadhanty Mutiah Lestari Mutiara Putri Mutiara Putri, Mutiara Nadiar Pertiwi Nadya Anggraini Nadya Anggraini Nasiroh, Anisa Qoriq Atun Nazhifah, Rizqia Nely Andriani Nely andriani Nely Andriani, Nely Nestity, Alifia Nia Nopeliza, Nia Nia Three Manurung Nilam Cahyati Nita Arrum Sari Nora Herliza Noviana, RA Sania Nur Kayla Lekati Nurhikmah, Putri Nurjana, Elsa Vini Eka Nurkholisa Fajriah Oktasari, Tina Patriot, Evelina Astra Pertiwi, Nadiar Prajaska, Nicola Pratiwi Ineke Anwar Putra, Gede Mudita Edi Putra, Guruh Sukarno Putri Maulida, Nabila Putri Wijayanti Putri, Arselly Rahmanda Putri, Jamiatul Khairunnisa Putri, Mutiara Juliana Putriyani, Fena Siska Qathrunnada, Sausan Rahma Hafitri Rahman, Nor Farahwahidah Abdul Rahmannisa, Amanda Ramadhani, Neysya Ditha Ramadhona, Tyse Rara Rara Rara, Rara Redondo, Fernando Eric Ribkoh, Ribkoh Risa Albeta Rita Inderawati Rizki Novianti, Rizki Romadoni, Muhammad Salmah Rianti Salmah Rianti Salsabillah, Zahwa Nur Saparini Saparini Saparini Sardianto Markos Siahaan Sardianto Markos Siahaan, Sardianto Markos Sari, Dwita Kartika Sari, Isma Sari, Mona Sekarani, Tresnane Suandti Seprina, Iin Sholeha, Nurlita Dwi Siahaan, Sardianto Marcos Siregar, Anpo Ferdy Siti Kurdiah Siti Nur Azizah Sopia Sopia Sri Mindia Vanessa Pratiwi Sri Zakiyah Sudirman Sudirman Sudirman Sudirman Suhadi Suhadi Suhadi Suhadi Suhadi Supardi Supardi Supardi Supari Supari Supari Supari Supari Supari Supari, Supari Sutanto, Taufiq Sutinah, Eka Syahputri, Selvia Mariska Syahrani, Revanola Gusti Syuhendri, Syuhendri Tamara Shelvy Maharani Tanti Febrianti Taufiq Taufiq Taufiq Taufiq Tine Aprianti Tita Ratna Wulan Dari Tresnane Suandti Sekarani Trista Meilanie Tsabita Husna Tyse Ramadhona Utami, Amanda Kurnia Vania, Amanda Asti VIOLANTI ANARKY, VIOLANTI Wahyuningsi, Wahyuningsi Wailaina Wardah Amalia Wati, Lira Diska Winarni, Kiki Ayu Wulandari, Dwi Noviana Yendri Tri Hartati Napitupulu Yenny Anwar Yesi Monalika Yuli Setianingsih Yuliana, Rahma Zahlia, Marsela Zahra Ayanna Firmansyah Zulherman Zulherman Zulherman Zulherman, Zulherman