Can the number be split into powers of 2? Yesterday while playing with my kid I noticed the number in his toy train:



So we have $$4281$$ that can be split into $$4-2-8-1$$ or $$2^2-2^1-2^3-2^0$$

So simple challenge: given a non-negative number as input, return consistent truthy and falsey values that represent whether or not the string representation of the number (in base 10 and without leading zeroes) can be somehow split into numbers that are powers of 2.

Examples:

4281      truthy (4-2-8-1)
164       truthy (16-4 or 1-64)
8192      truthy (the number itself is a power of 2)
81024     truthy (8-1024 or 8-1-02-4)
101       truthy (1-01)
0         falsey (0 cannot be represented as 2^x for any x)
1         truthy
3         falsey
234789    falsey
256323    falsey (we have 256 and 32 but then 3)
8132      truthy (8-1-32)

Tests for very large numbers (not really necessary to be handled by your code):
81024256641116  truthy (8-1024-256-64-1-1-16)
64512819237913  falsey


This is code-golf, so may the shortest code for each language win!
05AB1E, 9 8 bytes
Ýos.œåPZ

-1 byte thanks to @Emigna by using Z (max) for the list of 0s and 1s to mimic an any command for 1 (truthy).
Try it online or verify all test cases. (NOTE: The т in the header is 100 to only get the first 100 power of 2 numbers, instead of the first input amount of power of 2 numbers. It works with the input amount of power of 2 as well, but is pretty inefficient and might timeout on TIO if the input is large enough.)
Explanation:

Ý            # Create a list in the range [0,n], where n is the (implicit) input
             # (or 100 in the TIO)
             #  i.e. 81024 → [0,1,2,3,...,81024]
 o           # Raise 2 to the `i`'th power for each `i` in the list
             #  → [1,2,4,8,...,451..216 (nr with 24391 digits)]
  s          # Swap to take the input
   .œ        # Create each possible partition of this input
             #  i.e. 81024 → [[&quot;8&quot;,&quot;1&quot;,&quot;0&quot;,&quot;2&quot;,&quot;4&quot;],[&quot;8&quot;,&quot;1&quot;,&quot;0&quot;,&quot;24&quot;],...,[&quot;8102&quot;,&quot;4&quot;],[&quot;81024&quot;]]
     å       # Check for each if it's in the list of powers of 2
             #  → [[1,1,0,1,1],[1,1,0,0],...,[0,1],[0]]
      P      # Check for each inner list whether all are truthy
             #  → [0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0]
       Z     # Take the maximum (and output implicitly)
             #  → 1 (truthy)