Fetching the paper…
Reading the bibliography…
Deep learning is a powerful tool for solving nonlinear differential equations, but usually, only the solution corresponding to the flattest local minimizer can be found due to the implicit regularization of stochastic gradient descent.
On a new method of numerical solution of systems of nonlinear equations
D. F. Davidenko · 1953
Earlier work this paper cites.
Rounding errors in algebraic processes
J. H. Wilkinson · 1963
Earlier work this paper cites.
Deflation techniques for the calculation of further solutions of a nonlinear system
K. M. Brown and W. B. Gearhart · 1971
Earlier work this paper cites.
Widely convergent method for finding multiple solutions of simultaneous nonlinear equations
F. H. Branin · 1972
Earlier work this paper cites.
A systematic search method for obtaining multiple solutions of simultaneous nonlinear equations
K.-S. Chao, D.-K. Liu, and C.-T. Pan · 1975
Earlier work this paper cites.
Self‐Organization in Nonequilibrium Systems
G. Nicolis and I. Prigogine · 1977
Earlier work this paper cites.
Searching for multiple solutions of nonlinear systems
M.-J. Chien · 1979
Earlier work this paper cites.
Unique and multiple solutions of a family of differential equations modeling chemical reactions
L. R. Williams and R. W. Leggett · 1982
Earlier work this paper cites.
On a Painlevé-type boundary-value problem
P. Holmes and D. Spence · 1984
Earlier work this paper cites.
Coefficient-parameter polynomial continuation
A. J. Sommese A. P. Morgan · 1989
Earlier work this paper cites.
On some conjectures of turcotte, spence, bau, and holmes
S. P. Hastings and W. C. Troy · 1989
Earlier work this paper cites.
Efficient training of the backpropagation network by solving a system of stiff ordinary differential equations
Owens and Filkin · 1989
Earlier work this paper cites.
Neural algorithm for solving differential equations
Hyuk Lee and In Seok Kang · 1990
Earlier work this paper cites.
Continuation and path following
E. L. Allgower and K. Georg · 1993
Earlier work this paper cites.
Universal approximation bounds for superpositions of a sigmoidal function
A. R. Barron · 1993
Earlier work this paper cites.
Complex patterns in a simple system
J. E. Pearson · 1993
Earlier work this paper cites.
Neural-network-based approximations for solving partial differential equations
M. W. M. G. Dissanayake and N. Phan-Thien · 1994
Earlier work this paper cites.
Analog cellular neural network with application to partial differential equations with variable mesh-size
D. Gobovic and M. E. Zaghloul · 1994
Earlier work this paper cites.
A separating surface for the Painlevé differential equation
V. A. Noonburg · 1995
Earlier work this paper cites.
Engineering and economic applications of complementarity problems
M. C. Ferris and J. S. Pang · 1997
Earlier work this paper cites.
Artificial neural networks for solving ordinary and partial differential equations
I. E. Lagaris, A. Likas, and D. I. Fotiadis · 1998
Earlier work this paper cites.
Multiple solutions for 2mth order sturm–liouville boundary value problems
C. J. Chyan and J. Henderson · 2000
Earlier work this paper cites.
Multiplicity of positive solutions for higher order sturm-liouville problems
J. M. Davis, L. H. Erbe, and J. Henderson · 2001
Earlier work this paper cites.
Introduction to Numerical Continuation Methods
E. L. Allgower and K. Georg · 2003
Earlier work this paper cites.
Multiple stable periodic solutions in a model for hormonal control of the menstrual cycle
L. H. Clark, P. M. Schlosser, and J. F. Selgrade · 2003
Earlier work this paper cites.
Multiple positive solutions of a boundary value problem for ordinary differential equations
J. R. Graef, C. Qian, and B. Yang · 2003
Earlier work this paper cites.
A three point boundary value problem for nonlinear fourth order differential equations
J. R. Graef, C. Qian, and B. Yang · 2003
Earlier work this paper cites.
Theory and Applications of Fractional Differential Equations
A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo · 2006
Earlier work this paper cites.
Understanding the difficulty of training deep feedforward neural networks
X. Glorot and Y. Bengio · 2010
Earlier work this paper cites.
Homotopy based solutions of the navier-stokes equations for a porous channel with orthogonally moving walls
H. Xu, Z. Lin, S. Liao, J. Wu, and J. Majdalani · 2010
Cited alongside, same era.
Adaptive subgradient methods for online learning and stochastic optimization
J. Duchi, E. Hazan, and Y. Singer · 2011
Cited alongside, same era.
A numerical methodology for the Painlevé equations
B. Fornberg and J.A.C. Weideman · 2011
Cited alongside, same era.
Homotopy Analysis Method in Nonlinear Differential Equations
S. Liao · 2012
Cited alongside, same era.
A bootstrapping approach for computing multiple solutions of differential equations
W. Hao, J. D. Hauenstein, B. Hu, and A. J. Sommese · 2014
Cited alongside, same era.
D. Lei, Z. Sun, Y. Xiao, and W. Y. Wang · 2018
Later among the works it cites.
Two-level spectral methods for nonlinear elliptic equations with multiple solutions
Y. Wang, W. Hao, and G. Lin · 2018
Later among the works it cites.
A phase shift deep neural network for high frequency approximation and wave problems
W. Cai, X. Li, and L. Liu · 2019
Later among the works it cites.
PhaseDNN - a parallel phase shift deep neural network for adaptive wideband learning
W. Cai, X. Li, and L. Liu · 2019
Later among the works it cites.
Multi-scale deep neural networks for solving high dimensional PDEs
alphaXiv searches the wider corpus for related work and actual follow-ups.
alphaXiv is searching for related work…
D. P. Kingma and J. Ba · 2014
Cited alongside, same era.
Distinct solutions of finite-dimensional complementarity problems
M. Croci and P. E. Farrell · 2015
Cited alongside, same era.
Deflation techniques for finding distinct solutions of nonlinear partial differential equations
P. E. Farrell, Á. Birkisson, and S. W. Funke · 2015
Cited alongside, same era.
Delving deep into rectifiers: Surpassing human-level performance on imagenet classification
K. He, X. Zhang, S. Ren, and J. Sun · 2015
Cited alongside, same era.
Regularity theory and high order numerical methods for the (1d)-Fractional Laplacian
G. Acosta, J.P. Borthagaray, O. Bruno, and M. Maas · 2016
Cited alongside, same era.
A deflation technique for detecting multiple liquid crystal equilibrium states
J. H. Adler, D. B. Emerson, P. E. Farrell, and S. P. MacLachlan · 2016
Cited alongside, same era.
Deep Learning
I. Goodfellow, Y. Bengio, and A. Courville · 2016
Cited alongside, same era.
W. Cai and Z. J. Xu · 2019
Later among the works it cites.
How much over-parameterization is sufficient to learn deep relu networks?
Z. Chen, Y. Cao, D. Zou, and Q. Gu · 2019
Later among the works it cites.
A proof that rectified deep neural networks overcome the curse of dimensionality in the numerical approximation of semilinear heat equations
M. Hutzenthaler, A. Jentzen, Th. Kruse, and T. A. Nguyen · 2019
Later among the works it cites.
Deep Nitsche method: Deep Ritz method with essential boundary conditions
Y. Liao and P. Ming · 2019
Later among the works it cites.
Theory of the frequency principle for general deep neural networks
T. Luo, Z. Ma, Z. J. Xu, and Y. Zhang · 2019
Later among the works it cites.
Deep ReLU networks overcome the curse of dimensionality for bandlimited functions
H. Montanelli, H. Yang, and Q. Du · 2019
Later among the works it cites.
Exponential relu dnn expression of holomorphic maps in high dimension
J. A. A. Opschoor, C. Schwab, and J. Zech · 2019
Later among the works it cites.
Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations
M. Raissi, P. Perdikaris, and G. E. Karniadakis · 2019
Later among the works it cites.
On the convergence of Adam and beyond
S. J. Reddi, S. Kale, and S. Kumar · 2019
Later among the works it cites.
Training behavior of deep neural network in frequency domain
Z. J. Xu, Y. Zhang, and Y. Xiao · 2019
Later among the works it cites.
The phase diagram of approximation rates for deep neural networks
D. Yarotsky and A. Zhevnerchuk · 2019
Later among the works it cites.
A convergence theory for deep learning via over-parameterization
Z. Song Z. A.-Zhu, Y. Li · 2019
Later among the works it cites.
Explicitizing an implicit bias of the frequency principle in two-layer neural networks
Y. Zhang, Z. J. Xu, T. Luo, and Z. Ma · 2019
Later among the works it cites.
A type of generalization error induced by initialization in deep neural networks
Y. Zhang, Z. J. Xu, T. Luo, and Z. Ma · 2019
Later among the works it cites.
Towards understanding the spectral bias of deep learning, 2020
Yuan Cao, Zhiying Fang, Yue Wu, Ding-Xuan Zhou, and Quanquan Gu · 2020
Closest in time.
SelectNet: Self-paced learning for high-dimensional partial differential equations
Y. Gu, H. Yang, and C. Zhou · 2020
Closest in time.
Int-deep: A deep learning initialized iterative method for nonlinear problems
Jianguo Huang, Haoqin Wang, and Haizhao Yang · 2020
Closest in time.
Deep network approximation for smooth functions
J. Lu, Z. Shen, H. Yang, and S. Zhang · 2020
Closest in time.
Two-layer neural networks for partial differential equations: Optimization and generalization theory
T. Luo and H. Yang · 2020
Closest in time.
Error bounds for deep relu networks using the kolmogorov–arnold superposition theorem
H. Montanelli and H. Yang · 2020
Closest in time.
Physics-informed probabilistic learning of linear embeddings of nonlinear dynamics with guaranteed stability
S. Pan and K. Duraisamy · 2020
Closest in time.
When and why pinns fail to train: A neural tangent kernel perspective
Sifan Wang, Xinling Yu, and Paris Perdikaris · 2020
Closest in time.
A physics-informed neural network for wind turbine main bearing fatigue
Y. A. Yucesan and F. A. C. Viana · 2020
Closest in time.
Weak adversarial networks for high-dimensional partial differential equations
Y. Zang, G. Bao, X. Ye, and H. Zhou · 2020
Closest in time.