PHYDISCOVER V4 IDEA

[coolguy173]

V3 is still mostly symbolic regression. V4 should move toward an actual hypothesis-discovery system.

1. Discover relationships across multiple variables
   Instead of mainly searching:

A = f(x, y)
V4 should search for implicit relationships:

F(x, y, z, A, B) ≈ 0
For example, it could discover:

A / (x*y) - 2.75 ≈ 0
without choosing A beforehand.

2. Automatically discover constants
   If it finds:

A / (x*y) ≈ 2.7498
V4 should recognize:

2.7498 ≈ 2.75
and report:

A ≈ 2.75xy
It should distinguish learned constants from constants already present in the grammar.

3. Search for transformations
   Give it:

x
y
z
A
B
C
and let it transform variables:

x²
sqrt(x)
log(x)
1/x
x/y
x*y
x²/y
...
Then search combinations of those transformed quantities.

This is important because many interesting relationships aren't obvious in their original coordinates.

4. Discover scaling laws
   V4 should specifically look for:

Y
∝
X
n
and automatically estimate
n
.

For example, if the hidden data follows:

Y

7.3
X
1.73
it should be able to discover something like:

Y ∝ X^1.73
rather than only finding a complicated tree approximating it.

5. Discover dimensionless combinations
   This would be one of the biggest upgrades.

If variables have dimensions, V4 can search combinations such as:

A * B^2 / C
and ask whether they are approximately constant.

That lets it find relationships that are independent of units.

For physics-style discovery, this is extremely useful.

6. Multiple independent experiments
   Instead of one dataset:

experiment_1
V4 should accept:

experiment_1
experiment_2
experiment_3
...
and reward hypotheses that work across all experiments.

A relationship that only works in one dataset should be heavily penalized.

This helps prevent overfitting.

7. Noise resistance
   Real measurements won't be perfect.

V4 should use:

robust loss
median error
outlier resistance
bootstrap validation
train/test/validation datasets
So something like:

Y = 3.0002*x² + noise
can still result in:

Y ≈ 3x²
rather than producing an enormous ugly expression to fit every noisy point.

8. Equation equivalence detection
   This is very important.

V4 should understand that these are basically the same discovery:

A = BC
A/B = C
A/(BC) = 1
A - B*C = 0
V3 mostly sees different expression trees.

V4 should normalize them into the same underlying relationship.

9. Novelty engine
   The archive should classify discoveries:

KNOWN
DERIVED
REDUNDANT
NOVEL
For example:

A = x*y
A/x = y
→ REDUNDANT

But:

C = sqrt(x)*z
→ NOVEL

This prevents the archive from filling with hundreds of algebraically equivalent equations.

10. "Interestingness" score
    Every hypothesis could receive:

Accuracy 0.97
Simplicity 0.91
Generality 0.88
Novelty 0.94
Robustness 0.96

Interestingness 0.93
Then V4 can prioritize the most interesting discoveries instead of simply the lowest numerical error.

11. Active experimentation
    This is where V4 starts getting really interesting.

After finding:

Y ≈ 3X^1.7
the system should ask:

"What new measurements would best distinguish this hypothesis from competing hypotheses?"
It could propose:

X = 0.5
X = 1
X = 2
X = 5
X = 10
and determine which experiment would provide the most information.

That turns it from:

data → equation

into:

data → hypothesis → experiment → new data → improved hypothesis

12. Hidden-law challenge mode
    V4 should have a proper benchmark.

You secretly generate data from laws such as:

Y = 4.7x^1.83z
or:

Y = sqrt(x)*z^2 / 3.2
or:

Y = (x*y)/(z+2)
Then V4 receives only the observations.

At the end you compare:

TRUE LAW
↓
V4 DISCOVERY
↓
ERROR
↓
SIMPLICITY
↓
GENERALIZATION
This gives us an objective way to measure whether the system is actually improving.

The big V4 architecture
I'd make it roughly:

             RAW DATA
                │
                ▼
         DATA ANALYZER
                │
      ┌─────────┴─────────┐
      ▼                   ▼

VARIABLE TRANSFORMS DIMENSION ANALYZER
│ │
└─────────┬─────────┘
▼
HYPOTHESIS ENGINE
│
┌─────────┼─────────┐
▼ ▼ ▼
GP SEARCH POWER LAW INVARIANTS
│ │ │
└─────────┼─────────┘
▼
EQUIVALENCE
DETECTION
│
▼
ROBUST TESTING
│
▼
NOVELTY ANALYZER
│
▼
DISCOVERY ARCHIVE
│
▼
EXPERIMENT DESIGN
│
▼
NEW DATA
And most importantly:
V4 should not know that the answer is supposed to be physics.

It should be capable of discovering arbitrary mathematical structure.

Give it:

x1 x2 x3 y1 y2 y3 ...
and its job is simply:

Find simple, accurate, robust, non-redundant mathematical relationships hidden in the observations.
Then physics becomes one application of the engine rather than something hardcoded into it.

@Sam3360

[Sam3360]

working on it!