ERJ170 flight crew reported Flight Director and Auto Throttle systems failures on climb. The crew diverted to make a precautionary landing.

Narrative

On the ZZZ1 ZZZZZ1 departure in the vicinity of ZZZZZ2 a standard call out for VNAV was made by the flying pilot (myself). Pilot monitoring (Captain) pressed VNAV. All was normal until several seconds later the flight director bug started descending. Pilot Monitoring (PM) called for Pilot Flying (PF) to shallow out climb and follow Flight Director (FD). Over the next few seconds the FD moved below the horizon down to about ten degrees nose down. PM called for controls and took over as PF. I became PM at that point. The airspeed began to drop significantly at this point and PM First Officer (FO) noticed that auto throttles were at idle. PM called out 'throttles at idle;' at which point PF pushed the throttles to TOGA position. Auto throttles were not disengaged at this point. Once airspeed had come back up; PF let the auto throttles engage once more. Once engaged; the auto throttles went back to idle detention; at which point PF disengaged auto throttles and flew manually. Flight Director was turned off and ATC was notified of the issue. ATC provided vectors at this point and manual modes were selected. Several minutes later the decision was made to divert to ZZZ2. After several minutes on vectors; the FD was engaged and the issue seemed to be resolved. The ILS to Runway XXR at ZZZ2 was flown with the FD and FMS functioning normally. Both pilots discussed the nature of the situation and attempted to determine any potential human errors that may have contributed to this event. The flight plan; including all way points; altitudes; and airspeeds; were verified by each pilot prior to takeoff. We have been unable to account for any problems regarding the flight data entry into the FMS at this point. Presently; maintenance is inspecting the aircraft for potential problems that may have contributed.