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Cited by 117

The DOI matches its Crossref record; 74 of the 74 references it cites were found in a registry or an index.

Solving differential equations with neural networks: Applications to the calculation of cosmological phase transitions

Piscopo, Maria Laura, Spannowsky, Michael, Waite, Philip·2019·Physical Review D
DOI 10.1103/physrevd.100.016002
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Abstract

Starting from the observation that artificial neural networks are uniquely suited to solving optimization problems, and most physics problems can be cast as an optimization task, we introduce a novel way of finding a numerical solution to wide classes of differential equations. We find our approach to be very flexible and stable without relying on trial solutions, and applicable to ordinary, partial and coupled differential equations. We apply our method to the calculation of tunneling profiles for cosmological phase transitions, which is a problem of relevance for baryogenesis and stochastic gravitational wave spectra. Comparing our solutions with publicly available codes which use numerical methods optimized for the calculation of tunneling profiles, we find our approach to provide at least as accurate results as these dedicated differential equation solvers, and for some parameter choices, even more accurate and reliable solutions. In particular, we compare the neural network approach with two publicly available profile solvers, CosmoTransitions and BubbleProfiler, and give explicit examples where the neural network approach finds the correct solution while dedicated solvers do not. We point out that this approach of using artificial neural networks to solve equations is viable for any problem that can be cast into the form $\mathcal{F}(\stackrel{\ensuremath{\rightarrow}}{x})=0$, and is thus applicable to various other problems in perturbative and nonperturbative quantum field theory.

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Crossref · 10.1103/physrevd.100.016002

Cites

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Figures

FIG. 1 The upper panel shows the solutions to the first and second order ODEs of Eqs. (4) and (5), with boundary conditions as outlined in the text. The middle panel shows the numerical difference between the analytic solution and the NN predicted solution for both cases. The lower panel shows the differential contribution ˆ F to the loss across the entire domain.
FIG. 1 The upper panel shows the solutions to the first and second order ODEs of Eqs. (4) and (5), with boundary conditions as outlined in the text. The middle panel shows the numerical difference between the analytic solution and the NN predicted solution for both cases. The lower panel shows the differential contribution ˆ F to the loss across the entire domain.CC-BY · From the PDF
FIG. 2 The upper panel shows the solutions for the functions ϕ 1 and ϕ 2 to the coupled differential equation of Eq. (6). The middle panel displays the numerical difference between the analytic solution and the NN predicted solution for ϕ 1 . The lower panel shows the differential contribution ˆ F to the loss across the entire domain, from the equation for ϕ 1 . The three NN curves in each panel correspond to the first, second and third iteration steps in the training of the network, with iterative increase of the training domain, as described in the text.
FIG. 2 The upper panel shows the solutions for the functions ϕ 1 and ϕ 2 to the coupled differential equation of Eq. (6). The middle panel displays the numerical difference between the analytic solution and the NN predicted solution for ϕ 1 . The lower panel shows the differential contribution ˆ F to the loss across the entire domain, from the equation for ϕ 1 . The three NN curves in each panel correspond to the first, second and third iteration steps in the training of the network, with iterative increase of the training domain, as described in the text.CC-BY · From the PDF
FIG. 3 Numerical difference between the analytic solution and the NN predicted solution of Eq. (8), with boundary conditions as given in Eq. (9), over the domain ð x; y Þ ∈ ½ 0 ; 1 Š × ½ 0 ; 1 Š .
FIG. 3 Numerical difference between the analytic solution and the NN predicted solution of Eq. (8), with boundary conditions as given in Eq. (9), over the domain ð x; y Þ ∈ ½ 0 ; 1 Š × ½ 0 ; 1 Š .CC-BY · From the PDF
FIG. 4 Plot of the potential in Eq. (18), with λ ¼ α ¼ 1 , for the thick-wall (blue solid) and the thin-wall (red dashed) cases. For the latter, the position of the global minimum is also marked by the black dot for clarity.
FIG. 4 Plot of the potential in Eq. (18), with λ ¼ α ¼ 1 , for the thick-wall (blue solid) and the thin-wall (red dashed) cases. For the latter, the position of the global minimum is also marked by the black dot for clarity.CC-BY · From the PDF
FIG. 5 The upper panel shows the bubble profile for the thickwall potential ( ϵ ¼ 0 . 3 ) in Eq. (18) for one scalar field, as obtained by our NN method, BubbleProfiler and CosmoTransitions. The middle panel displays the numerical difference between the NN predicted solution and the solutions from the other two codes. The lower panel shows the differential contribution ˆ F to the loss.
FIG. 5 The upper panel shows the bubble profile for the thickwall potential ( ϵ ¼ 0 . 3 ) in Eq. (18) for one scalar field, as obtained by our NN method, BubbleProfiler and CosmoTransitions. The middle panel displays the numerical difference between the NN predicted solution and the solutions from the other two codes. The lower panel shows the differential contribution ˆ F to the loss.CC-BY · From the PDF
FIG. 6 The upper panel shows the bubble profile for the thinwall potential ( ϵ ¼ 0 . 01 ) in Eq. (18) for one scalar field, as obtained by our NN method, BubbleProfiler and CosmoTransitions. The middle panel displays the numerical difference between the NN predicted solution and the solutions from the other two codes. The lower panel shows the differential contribution ˆ F to the loss. The dotted vertical line shows the analytic location of the bubble radius, which agrees with the radius found by both the NN and BubbleProfiler.
FIG. 6 The upper panel shows the bubble profile for the thinwall potential ( ϵ ¼ 0 . 01 ) in Eq. (18) for one scalar field, as obtained by our NN method, BubbleProfiler and CosmoTransitions. The middle panel displays the numerical difference between the NN predicted solution and the solutions from the other two codes. The lower panel shows the differential contribution ˆ F to the loss. The dotted vertical line shows the analytic location of the bubble radius, which agrees with the radius found by both the NN and BubbleProfiler.CC-BY · From the PDF
FIG. 7 The upper panel shows the bubble profiles for the thickwall potential in Eq. (20) for two scalar fields, as obtained by our NN method, BubbleProfiler and CosmoTransitions. The middle panel displays the numerical difference between the NN predicted solutions and the solutions from the other two codes. The lower panel shows the differential contribution ˆ F to the loss from ϕ 1 and ϕ 2 .
FIG. 7 The upper panel shows the bubble profiles for the thickwall potential in Eq. (20) for two scalar fields, as obtained by our NN method, BubbleProfiler and CosmoTransitions. The middle panel displays the numerical difference between the NN predicted solutions and the solutions from the other two codes. The lower panel shows the differential contribution ˆ F to the loss from ϕ 1 and ϕ 2 .CC-BY · From the PDF
FIG. 8 Calculated solutions for the tunneling path for NN, BubbleProfiler and CosmoTransitions. The paths range from the local minimumtotheexit point of the tunneling barrier. Also shown are the contours of the potential, where the global minimum is denoted by the black dot and the local minimum by the black cross.
FIG. 8 Calculated solutions for the tunneling path for NN, BubbleProfiler and CosmoTransitions. The paths range from the local minimumtotheexit point of the tunneling barrier. Also shown are the contours of the potential, where the global minimum is denoted by the black dot and the local minimum by the black cross.CC-BY · From the PDF
FIG. 9 The upper panel shows the bubble profiles for the singlet-scalar extended Standard Model potential in Eq. (26), as obtained by our NN method, BubbleProfiler and CosmoTransitions. The middle panel displays the numerical difference between the NN predicted solutions and the solutions from the other two codes. The lower panel shows the differential contribution ˆ F to the loss from h and s .
FIG. 9 The upper panel shows the bubble profiles for the singlet-scalar extended Standard Model potential in Eq. (26), as obtained by our NN method, BubbleProfiler and CosmoTransitions. The middle panel displays the numerical difference between the NN predicted solutions and the solutions from the other two codes. The lower panel shows the differential contribution ˆ F to the loss from h and s .CC-BY · From the PDF

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Cites

Research works printed but not linked: 3

  1. M. E. Shaposhnikov, 1986, JETP Lett., 44, 465
  2. A. D. Linde, 1990, Contemp. Concepts Phys., 5, 1
  3. L. V. Kantorovich, 1948, Usp. Math. Nauk, 3, 89

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Metadata

Authors
Piscopo, M. L., Spannowsky, M., & Waite, P.
ORCID
Waite, Philip
Venue
Physical Review D
Publisher
American Physical Society (APS)
Published
2019-07-09
Volume
100
Issue
1
ISSN
  • 2470-0010
  • 2470-0029
DOI
10.1103/physrevd.100.016002
Cited by
117
References
74
Language
en
Licence
creativecommons.org/licenses/by/4.0/ (vor)
Funders
  • University of Tuebingen
  • Humboldt Society

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