I struggled with the same problem just a minute ago. My project was generated using the v 1.6.0 of angular-cli.

when I am going to start my ng server, then encounter the below error.

ng s

error⬇︎

module.js:557
    throw err;
    ^
Error: Cannot find module '@angular-devkit/core'
    at Function.Module._resolveFilename (module.js:555:15)
    at Function.Module._load (module.js:482:25)
    at Module.require (module.js:604:17)
    at require (internal/module.js:11:18)
    at Object.<anonymous> (/workspace/observable-app/node_modules/@angular-devkit/schematics/src/tree/virtual.js:10:16)
    at Module._compile (module.js:660:30)
    at Object.Module._extensions..js (module.js:671:10)
    at Module.load (module.js:573:32)
    at tryModuleLoad (module.js:513:12)
    at Function.Module._load (module.js:505:3)

do not worry, here is a solution.
first, you need to change the @angular/cli version.
editing my package.json changing the line

"@angular/cli": "1.6.0",
to
"@angular/cli": "^1.6.0",

then now

npm update

and it works.

I hope it will help you too.

Enjoy Coding...