Partial Differential Equation
A mathematical equation describing how a physical quantity varies with multiple independent variables such as time and space, involving an unknown function and its partial derivatives. It is widely used in engineering simulation in fluid mechanics, heat conduction, and electromagnetics, and is the primary application scenario for operator surrogate models.
Timeline (last 90 days)
Terence Tao has long worked in areas such as harmonic analysis, number theory, and partial differential equations.
The traditional computational mathematics workflow for solving PDEs includes geometric modeling and mesh generation, selection of discretization schemes, handling of boundary conditions, time stepping or iterative solving, and post-processing and error analysis.
Deep learning methods compress the PDE-solving workflow into defining a network, constructing a loss, and optimizing training, but they shift the difficulty to optimization and generalization; network architecture, activation functions, loss weights, and sampling strategies all require repeated trial and error and lack systematic design guidelines.
Numerical solutions of partial differential equations often require errors on the order of 10^-3 to 10^-4, whereas deep learning for solving PDEs lacks a numerical analysis process, so the quality and usability of results often cannot be judged.
Gaussian process (GP) solvers return a posterior distribution and can quantify solution uncertainty, rather than merely providing a point estimate as traditional numerical methods do.
The Navier-Stokes problem involves the core proposition of global regularity for nonlinear equations in the theory of partial differential equations
The RealPDE competition is a competition under the NeurIPS 2026 framework, focusing on the solving of partial differential equations (PDEs) in scientific machine learning and alignment with real-world data.
All Facts (7)
Terence Tao has long worked in areas such as harmonic analysis, number theory, and partial differential equations.
50%UnverifiedThe traditional computational mathematics workflow for solving PDEs includes geometric modeling and mesh generation, selection of discretization schemes, handling of boundary conditions, time stepping or iterative solving, and post-processing and error analysis.
50%UnverifiedDeep learning methods compress the PDE-solving workflow into defining a network, constructing a loss, and optimizing training, but they shift the difficulty to optimization and generalization; network architecture, activation functions, loss weights, and sampling strategies all require repeated trial and error and lack systematic design guidelines.
50%UnverifiedNumerical solutions of partial differential equations often require errors on the order of 10^-3 to 10^-4, whereas deep learning for solving PDEs lacks a numerical analysis process, so the quality and usability of results often cannot be judged.
50%UnverifiedGaussian process (GP) solvers return a posterior distribution and can quantify solution uncertainty, rather than merely providing a point estimate as traditional numerical methods do.
50%UnverifiedThe Navier-Stokes problem involves the core proposition of global regularity for nonlinear equations in the theory of partial differential equations
50%UnverifiedThe RealPDE competition is a competition under the NeurIPS 2026 framework, focusing on the solving of partial differential equations (PDEs) in scientific machine learning and alignment with real-world data.
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