Understanding Thermal Stability and Drift in Linear Regulators
One option is to close the loop around the regulator:



The capacitors are necessary only due to the part I used in simulation - the parts would need choosing carefully for your application to ensure stability.

The reference voltage should be a bandgap type device and the feedback resistors may need to be tight tolerance (although a multi-turn adjustable might be suitable).

What happens here is quite straightforward: As Vout decreases, the loop drives such that more current is forced through R2. The loop control range is limited (and I have not yet analysed that).

Another option (as noted in the comments) is to use a digipot; I have used this part in the past quite successfully but that was not a precision circuit.

The absolute accuracy is not particularly good, but as part of an overall feedback control loop can be quite accurate.

The tolerance build-up of the measurement circuit will need to be addressed as the INL, DNL and Effective Number Of Bits in particular all add to the system error budget.

Whether tiny errors in the range of ppm (dynamically) are achievable with that setup is something I have not yet analysed. A true 20 bit conversion can achieve just under 1ppm accuracy, but whether that can be achieved on a PCB depends on many things.

To get a 20 bit effective conversion would first require a 24 bit conversion; there are a number of techniques around that help with ADC performance, such as dithering and noise reduction techniques.