Solves Fu-Yau equation for negative slope parameters in arbitrary dimensions.
problem Solving the Fu-Yau equation for negative slope parameters in arbitrary dimensions.
method Solves the Fu-Yau equation for negative slope parameters in arbitrary dimensions.
result First non-trivial solutions of the Fu-Yau equation in any dimension strictly greater than 2.
Solves critical LYZ equation in Kähler geometry.
problem Solvability of LYZ equation at critical phase.
method Establishes existence of smooth solutions.
result Solves critical case of LYZ equation.
Develops ML method for solving financial equations.
problem Solving financial equations efficiently and accurately.
method Combines semi-analytical and numerical techniques.
result Significantly faster and more accurate solutions.
RNN operators solve Newton's equations with large timesteps for molecular dynamics.
problem Solving Newton's equations of motion with large timesteps for molecular dynamics simulations.
method Recurrent Neural Networks (RNN) operators to solve Newton's equations using past trajectory data.
result Significant speedup in molecular dynamics simulations with timesteps up to 4000 times larger.
Solved metrisability for Painlevé equations and their geodesics.
problem Metrisability of Painlevé equations and geodesics.
method Solved the metrisability problem for all 2nd order ODEs with Painlevé property.
result Integral curves of Painlevé equations are geodesics of a (pseudo) Riemannian metric.
Quantum model discovery uses DQCs to solve equations from data.
problem Discovering differential equations from data using quantum computing.
method Differentiable quantum circuits (DQCs) to solve parameterized equations, regression on data and equations.
result Successful parameter inference and equation discovery on various systems.
Solves open problems for fully nonlinear elliptic equations on manifolds.
problem Solving fully nonlinear elliptic equations on manifolds.
method Analytic slope invariant and Nakai-Moishezon criterion.
result Solves open problems including hessian and hessian quotient equations.
Method solves inverse problems for semilinear equations with power nonlinearities.
problem Solving inverse problems for semilinear equations with power nonlinearities.
method Higher order linearizations based on a nonlinear Dirichlet-to-Neumann map.
result Solves inverse problems for certain semilinear equations in dimensions 2 and n≥3.
The paper proposes a different method of solving a simplified version of the Black-Scholes equation. This paper will discuss the importance of the Black-Scholes equation and its applications in finance.
Solves tangle equations involving composite links under specific conditions.
problem Solving tangle equations with composite links and 2-bridge links.
method Uses algebraic and double branched cover properties to solve equations.
result Non-hyperbolic solutions to tangle equations involving composite links are found.
Neural networks solve SPDEs using Wiener chaos expansion.
problem Solving stochastic partial differential equations (SPDEs) numerically.
method Using neural networks in the truncated Wiener chaos expansion.
result Approximation rates for learning SPDE solutions with noise.
Solves Einstein constraint equations on compact manifolds with specified boundaries.
problem Solving Einstein constraint equations with specified boundaries.
method Studies conformal constraint equations with low regularity assumptions.
result Solves Einstein constraint equations on compact manifolds with specified boundaries.
Solves CR Poincaré-Lelong equation on CR manifolds, revealing structures and solitons.
problem Solving CR Poincaré-Lelong equation on CR manifolds.
method Solves CR Poisson equation on CR (2n+1)-manifolds with specific curvature properties. result Discovers structures and CR Yamabe steady solitons on complete noncompact Sasakian manifolds.
Proposes a method to train neural networks that solve differential equations faster.
problem Training neural networks that solve differential equations becomes computationally expensive.
method Introduces a differentiable surrogate for numerical solver time cost using higher-order derivatives.
result Trains models that are faster to solve while maintaining nearly the same accuracy.
Solves a differential equation problem using Cartan's method.
problem Equivalence of second order ordinary differential equations under point transformations.
method E. Cartan's method of equivalence
result Solves the point equivalence problem for second order ODEs.
Neural networks can solve complex PDEs with minimal parameters.
problem Using neural networks to solve partial differential equations.
method Investigated two PDEs: Poisson and steady Navier--Stokes. Analyzed neural network architecture, initialization, loss function, and compared to classical methods.
result Small neural networks (<500 learnable parameters) can accurately solve complex PDEs.
New method solves PDEs for any initial condition without retraining.
problem Solving PDEs for different initial conditions requires retraining neural solvers.
method Formulate solution as conditional probability distribution.
result Approximates PDE solution for arbitrary initial conditions.
This paper proposes an unsupervised learning method to solve heat equations on chips.
problem Critical need for solving heat transfer equations on chips for 5G and AI.
method Hybrid framework of Auto Encoder and Image Gradient for unsupervised learning.
result Framework can solve heat transfer problems with a single training process and predict unseen cases.
New method solves elliptic equations on manifolds without grids.
problem Solving elliptic equations on complex manifolds.
method Numerical domain decomposition method avoiding global grids.
result Method validated on specific 4D manifolds.
Solves a specific Dirichlet problem on Hermitian manifolds.
problem Solving Dirichlet problem for Monge-Ampère type equations on Hermitian manifolds.
method Solves the Dirichlet problem for Monge-Ampère type equations for (n−1)-plurisubharmonic functions on Hermitian manifolds. result Solves a specific Dirichlet problem on Hermitian manifolds.
PyDEns framework uses neural nets to solve PDEs.
problem Lack of flexible framework for solving PDEs with neural networks.
method PyDEns-module coupled with BatchFlow, allowing to solve PDEs, search for neural network architectures, and control model training.
result Ready-to-use and open-source numerical solver of PDEs based on neural networks.
Paper solves a long-standing problem with curvature estimates.
problem Long-standing problem in n−2 curvature equation. method Global curvature estimate for the n−2 Hessian equation. result Solves a long-standing problem in n−2 curvature equation. New method solves tensor equations including parity odd and even terms in 4D.
problem Solving linear tensor equations with parity odd and even terms in 4D.
method Extending previous results, solving a 30-parameter linear tensor equation step by step.
result Explicit solution for tensor field components in terms of known components.
Solves generalized Kazdan-Warner equations on foliated manifolds.
problem Existence and uniqueness of solutions to generalized Kazdan-Warner equations on foliated manifolds.
method Extends theorem to compact foliated manifolds, provides examples of PDEs.
result Solves the transverse Hitchin equation and its generalizations.
Researchers solve field equations for special gravitational instantons.
problem Solving field equations for conformally Kähler Riemannian four-manifolds.
method Developed a framework to solve the field equations for generalised gravitational instantons using conformal self-duality and cosmological Einstein-Maxwell.
result Found conformally self-dual and Einstein-Maxwell generalisations of specific geometries.
We solve linear equations with tensors of any rank.
problem Solving linear equations involving tensors of arbitrary rank.
method Developed a systematic approach for tensors of rank 3 and generalized to arbitrary rank.
result Derived a solution for tensors of arbitrary rank.
Solves geodesic equations on specific metrics.
problem Explicit geodesic solutions for certain metrics are not known.
method Solves geodesic equations under different constant conditions.
result Explicit solutions not available for all conditions.
Solve-training trains neural nets to map physical solutions efficiently.
problem Representing complex physical solutions with neural networks.
method Variational training using loss functions from physical models.
result Effective neural network representation of solution maps without expensive labels.
Solves a specific Dirichlet problem for Lagrangian mean curvature equations.
problem Solving the Dirichlet problem for Lagrangian mean curvature equations.
method Solves the Dirichlet problem for Lagrangian mean curvature equations on uniformly convex domains.
result Solves the Dirichlet problem for Lagrangian mean curvature equations.
Deep reinforcement learning solves complex differential equations.
problem Solving nonlinear differential equations.
method Rule-based deep reinforcement learning approach.
result Solver captures intrinsic nature of equations with high accuracy.
Solves a Monge-Ampère type equation for Nakano positive curvature tensors of holomorphic vector bundles.
problem Solving Monge-Ampère type equations for Nakano positive curvature tensors of holomorphic vector bundles.
method Solves the Monge-Ampère type equation in the conformal class of a Nakano positive Hermitian metric.
result Solves the Monge-Ampère type equation for Nakano positive curvature tensors of holomorphic vector bundles.
The Poisson equation on manifolds plays an fundamental role in many applications. Recently, we proposed a novel numerical method called the Point Integral method (PIM) to solve the Poisson equations on manifolds from point clouds. In this paper, we prove the convergence of the point integral method for solving the Pois…
Neural network learns to solve Black-Scholes for stock options.
problem Stock option pricing using the Black-Scholes Equation.
method Neural Networks applied to solve the Black-Scholes Equation.
result Neural network can accurately forecast stock option prices.
DPINN improves data efficiency and accuracy in solving PDEs.
problem Solving partial differential equations efficiently and accurately.
method Proposed a distributed physics-informed neural network (DPINN) to improve upon the original PINN.
result DPINN yields more accurate and data-efficient solutions to PDEs.
Study solves HJB equations for time-inconsistent control problems.
problem Time-inconsistent deterministic linear quadratic control problems.
method Characterized solutions using Riccati equations with integral terms, proving uniqueness.
result Uniqueness of solutions to equilibrium HJB equations proved.
Solves geodesic equations on specific metrics types.
problem Finding explicit solutions for geodesic equations on Eguchi-Hanson metrics.
method Analyzes geodesic equations under different constant conditions.
result Explicit solutions not yet available for geodesic equations when only ψ is constant. Gradient-free learning uses kernel and range space for solving linear equations.
problem Solving linear equations and least squares problems.
method Manipulating kernel and range space to solve linear matrix equations, adapting for neural networks.
result Gradient-free learning framework for neural networks, showing good performance on real-world data.
Deep learning model solves high-dimensional PDEs using Actor-Critic approach.
problem Solving high-dimensional nonlinear PDEs efficiently.
method Reformulated PDE into BSDE system, inspired by Actor-Critic algorithm for deep RL.
result Improved model with fewer parameters, faster convergence, and less hyperparameter tuning.
Study SOLV geometry using monopole Floer homology.
problem Prove SOLV rational homology sphere Y is an L-space.
method Apply Fourier analysis on solvable groups to show irreducible solutions do not exist for certain metrics.
result Y is an L-space geometrically proven.
Study solves inverse problems for equations with fractional nonlinearities.
problem Solving inverse problems for semilinear elliptic equations with fractional power nonlinearities.
method Higher order linearization method adapted for fractional order.
result Results of previous studies remain valid for general power nonlinearities.
Deep neural nets solve complex insurance math equations.
problem Optimal control problems in insurance math.
method Deep neural network algorithm for elliptic PDEs.
result Solves high-dimensional semilinear elliptic PDEs.
Solves Jang equation for hyperboloidal data, proving positive mass theorem.
problem Proving the positive mass theorem in asymptotically hyperbolic 3D spacetimes.
method Solves Jang equation with hyperboloidal initial data, applies to positive mass theorem.
result Non-spinor proof of positive mass theorem in 3D asymptotically hyperbolic spacetimes.
Solves second-order PDEs using quotients and differential invariants.
problem Solving second-order PDEs with first-order quotients.
method Solve the quotient PDE using differential invariants, then add new constraints to solve the original PDE.
result New method for solving second-order scalar PDEs with infinite-dimensional symmetry algebras.
Paper generalizes sub-slope definition and solves complex equations on compact manifolds.
problem Solving complex equations on compact almost Hermitian manifolds.
method Generalized sub-slope definition and proved existence of solutions for a class of equations.
result Solved complex Hessian quotient and deformed Hermitian-Yang-Mills equations.
Goursat showed that in the presence of an intermediate integral, the problem of solving a second-order Monge-Ampere equation can be reduced to solving a first-order equation, in the sense that the generic solution of the first-order equation will also be a solution of the original equation. An attempt by Hermann to giv…
The paper solves equations for Higgs bundles on non-Kähler manifolds.
problem Analytically stable Higgs bundles on non-Kähler manifolds.
method Solving the Hermitian-Einstein equation on analytically stable Higgs bundles under specific conditions.
result Solutions to the Hermitian-Einstein equation for analytically stable Higgs bundles on non-Kähler manifolds.
New method solves robust matrix completion using nonlinear equations.
problem Recover low rank and sparse matrices from incomplete observations.
method Transforms problem into solving a system of nonlinear equations, then uses the alternative direction method.
result Algorithm converges linearly to the true solution under proper assumptions.
New machine learning method solves semi-linear PDEs more accurately.
problem Solving semi-linear partial differential equations (PDEs)
method Improves existing machine learning algorithms using neural networks and deep learning techniques.
result New algorithm outperforms existing methods in accuracy.