Table 1 Error comparison for simple harmonic oscillator.

From: A novel hybrid framework for efficient higher order ODE solvers using neural networks and block methods

Time

Neural-ODE Hybrid Block Method Error

Explicit Euler Method Error

Implicit Euler Method Error

Adams-Bashforth Method Error

BDF Method Error

Spectral Collocation Method Error

0.1

0.0000000003

0.0000004717

0.0000003723

0.0000000440

0.0000001626

0.0000000591

0.2

0.0000000007

0.0000001883

0.0000002596

0.0000004519

0.0000005052

0.0000000386

0.3

0.0000000005

0.0000004415

0.0000003338

0.0000000495

0.0000001360

0.0000000178

0.4

0.0000000008

0.0000005454

0.0000004650

0.0000001233

0.0000002478

0.0000001664

0.5

0.0000000009

0.0000002040

0.0000000970

0.0000002174

0.0000001511

0.0000002830

0.6

0.0000000002

0.0000001200

0.0000000998

0.0000002599

0.0000004611

0.0000002100

0.7

0.0000000009

0.0000002360

0.0000003403

0.0000002874

0.0000002037

0.0000008694

0.8

0.0000000004

0.0000002175

0.0000002967

0.0000001535

0.0000001013

0.0000001113

0.9

0.0000000006

0.0000002664

0.0000003935

0.0000001209

0.0000000289

0.0000010287

0.1

0.0000000003

0.0000004717

0.0000003723

0.0000000440

0.0000001626

0.0000000591

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