BEGIN:VCALENDAR VERSION:2.0 PRODID:-//wp-events-plugin.com//7.2.3.1//EN TZID:Asia/Kolkata X-WR-TIMEZONE:Asia/Kolkata BEGIN:VEVENT UID:152@cds.iisc.ac.in DTSTART;TZID=Asia/Kolkata:20251015T140000 DTEND;TZID=Asia/Kolkata:20251015T150000 DTSTAMP:20251007T043455Z URL:https://cds.iisc.ac.in/events/ph-d-thesis-colloquium-102-cds-15-octobe r-2025-transformer-neural-operators-for-learning-generalized-solutions-of- partial-differential-equations-and-data-assimilation/ SUMMARY:Ph.D: Thesis Colloquium: 102 : CDS: 15\, October 2025 "Transformer Neural Operators for Learning Generalized Solutions of Partial Differentia l Equations and Data Assimilation" DESCRIPTION:DEPARTMENT OF COMPUTATIONAL AND DATA SCIENCES\nPh.D. Thesis Col loquium\n\n\n\nSpeaker: Mr. Boya Sumanth Kumar\nS.R. Number: 06-18-01-10-1 2-20-1-18435\nTitle: "Transformer Neural Operators for Learning Generalize d Solutions of Partial Differential Equations and Data Assimilation"\nRese arch Supervisor: Dr. Deepak Subramani\nDate &\; Time : October 15\, 202 5 (Wednesday)\, 02:00 PM\nVenue : #102\, CDS Seminar Hall\n\n\n\nABSTRACT\ nScientific machine learning is transforming the way we solve physical dyn amics governed by partial differential equations (PDEs). Solving nonlinear PDEs across multiple initial and boundary conditions requires re-running traditional numerical solvers\, and existing physics-informed neural netwo rks require costly retraining for each new condition. Furthermore\, with t he availability of sparse observational data in the meteorological and oce an science domains\, the necessity for data assimilation in forward simula tions is crucial. The above challenges are addressed by proposing three ne ural operators with architectural innovations: (1) the Physics-Informed Tr ansformer Neural Operator (PINTO) for simulation-free PDE solving that is trained using only physics loss and generalizes to unseen initial and boun dary conditions\, (2) the Physics-Guided Transformer Neural Operator (PGNT O) for finding generalizable PDE solutions from simulation data in the abs ence of governing equations\, and (3) the Implicit Neural Transformer Oper ator (INTO) for the data assimilation task of reconstructing continuous fi elds from limited observational data. Collectively\, the above contributio ns push the boundaries of scientific machine learning in solving PDEs and data assimilation.\n\nThe first part of this thesis introduces a novel Phy sics-Informed Transformer Neural Operator (PINTO). Current neural operator approaches\, though capable of learning functional mappings between infin ite-dimensional spaces\, suffer from two critical limitations: dependence on substantial simulation data and poor generalization to unseen condition s. PINTO addresses these fundamental challenges by introducing a novel phy sics-informed framework that achieves efficient generalization through sim ulation-free\, physics-only training. The core innovation of PINTO lies in the development of iterative kernel integral operator units that leverage cross-attention mechanisms to transform domain points into initial and bo undary condition-aware representation vectors. This attention-based archit ecture enables context-aware learning that fundamentally differs from exis ting neural operators\, allowing PINTO to learn mappings from input functi ons (initial/boundary conditions) to complete PDE solution spaces through a single forward pass. The architecture comprises three key stages: liftin g layers for projecting the solution's domain coordinates to a higher-dime nsional representation space\, iterative kernel integration layers for con text-aware representation learning\, and projection layers for mapping the learned representation to the solution space. We demonstrate PINTO's supe rior performance across critical fluid mechanics and engineering applicati ons\, including linear advection\, nonlinear Burgers equation\, and steady and unsteady Navier-Stokes equations spanning multiple flow scenarios. Un der challenging unseen conditions\, PINTO achieves relative errors of mere ly 20 to 33% of those obtained by state-of-the-art physics-informed neural operator methods when compared with analytical and numerical solutions. C ritically\, PINTO exhibits temporal extrapolation capabilities absent in c ompeting approaches\, accurately solving the advection and Burger’s equa tions in time steps not present during training.\n\nIn the second part of the thesis\, we extend our framework to physics-guided transformer neural operators (PGNTO)\, trained on simulation data rather than physics loss\, to address scenarios where governing PDEs are unavailable and to eliminate the training instabilities that arise due to physics loss for high-dimens ional PDEs. We introduce Gabor filters in the coordinate lifting layers to develop PGNTO\, a version of our transformer neural operator that shows i mproved PDE solving with simulation data. Our PGNTO successfully solves st andard PDE benchmarks that include the nonlinear Burgers\, Darcy\, 2D Navi er Stokes\, plasticity\, elasticity\, and also scales to complex 3D turbul ent flows\, specifically predicting wake dynamics behind wind turbines und er varying inlet velocities. Across all test cases\, PGNTO maintains super ior accuracy with relative errors that are only one-third of competing neu ral operators.\n\nFinally\, in the third part of the thesis\, we introduce the implicit neural transformer operator (INTO) for neural data assimilat ion\, enabling continuous field reconstruction from sparse observational d ata. By modifying PINTO's architecture with Gabor filters for coordinate l ifting and Fourier filters for value representation\, INTO outperforms exi sting implicit neural representation networks on reconstruction tasks. Uni quely\, a single INTO model generalizes across varying sparsity levels\, e liminating the need for training separate models for different data availa bilities. The performance of INTO is demonstrated on global surface temper ature and air temperature datasets\, and compared with other leading deep neural models for physical field reconstruction.\n\nOverall\, this thesis establishes transformer-based neural operators as a unified framework for physics-informed PDE solving and neural data assimilation\, achieving unpr ecedented generalization\, computational efficiency\, and accuracy while r equiring minimal training data. Our suite of transformer neural operators opens new avenues for real-time prediction and data assimilation in physic s and engineering applications.\n\n\n\nALL ARE WELCOME CATEGORIES:Events,Ph.D. Thesis Colloquium END:VEVENT BEGIN:VTIMEZONE TZID:Asia/Kolkata X-LIC-LOCATION:Asia/Kolkata BEGIN:STANDARD DTSTART:20241015T140000 TZOFFSETFROM:+0530 TZOFFSETTO:+0530 TZNAME:IST END:STANDARD END:VTIMEZONE END:VCALENDAR