Using Prior Knowledge Ji Kubalk jiri.kubalik@cvut.cz Symbolic - - PowerPoint PPT Presentation

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Using Prior Knowledge Ji Kubalk jiri.kubalik@cvut.cz Symbolic - - PowerPoint PPT Presentation

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Symbolic Regression Using Prior Knowledge Ji Kubalk jiri.kubalik@cvut.cz Symbolic Regression Using Prior Knowledge Insufficient training data sparse and noisy, unevenly sample the input space, may completely omit some parts of

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Symbolic Regression Using Prior Knowledge

Jiří Kubalík

jiri.kubalik@cvut.cz

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CIIRC Meeting on genetic and related methods, 17 February 2020 [2]

Symbolic Regression Using Prior Knowledge

Insufficient training data

  • sparse and noisy,
  • unevenly sample the input space,
  • may completely omit some parts of the input space.

Models trained using only such training data tend to be

  • verfitted,
  • partially incorrect in terms of their steady-state characteristics or

local behavior.

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CIIRC Meeting on genetic and related methods, 17 February 2020 [3]

Magnetic manipulation

Magnetic manipulation – an iron ball moving along a rail and an electromagnet at a static position under the rail. Data – noisy; only a part of the input space is covered. Goal is to find a model of the nonlinear magnetic force affecting the ball as a function of the distance between the ball and the activated coil.

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CIIRC Meeting on genetic and related methods, 17 February 2020 [4]

Magman: SR driven by training data only

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CIIRC Meeting on genetic and related methods, 17 February 2020 [5]

Two resistors in parallel

Resistance – equivalent resistance of two resistors in parallel. Data – very sparse and noisy. Goal is to find a model that fits the data and obeys the physical law. Baseline model: 𝑆 =

𝑆1𝑆2 𝑆1+𝑆2

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CIIRC Meeting on genetic and related methods, 17 February 2020 [6]

Resistance: SR driven by training data only

Baseline model SR model

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CIIRC Meeting on genetic and related methods, 17 February 2020 [7]

Magman: Desired model’s properties

  • Increasing monotonicity

𝑦 ∈ (−0.075, −0.01)

  • r

𝑦 ∈ (0.01, 0.075)

  • Decreasing monotonicity

𝑦 ∈ (−0.01, 0.01)

  • Odd symmetry
  • Exact output values

𝑔 −0.075 = 0.001 𝑔 0.075 = −0.001, 𝑔 0 = 0.0

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CIIRC Meeting on genetic and related methods, 17 February 2020 [8]

Resistance: Desired model’s properties

  • symmetry with respect to arguments

R(R1, R2) = R(R2, R1)

  • domain-specific constraint

R1 = R2 ⇒ R(R1, R2) = R1 /2

  • domain-specific constraint

R(R1, R2) ≤ R1, R(R1, R2) ≤ R2

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CIIRC Meeting on genetic and related methods, 17 February 2020 [9]

Bi-objective Symbolic Regression

  • Optimisation criteria
  • minimise prediction error on training data samples
  • minimise violation of the desired model’s properties
  • Constraint samples set – properties are internally represented by a set of

discrete data samples on which candidate models are exactly checked.

  • NSGA-II – based on the concept of dominance
  • generates a set of non-dominated solutions
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CIIRC Meeting on genetic and related methods, 17 February 2020 [10]

Bi-objective SR: Magman

Accurate and valid Inaccurate, but perfectly valid

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CIIRC Meeting on genetic and related methods, 17 February 2020 [11]

Bi-objective SR: Resistors

SR model Baseline model

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CIIRC Meeting on genetic and related methods, 17 February 2020 [12]

Summary

  • Multi-objective SR method that produces realistic models that fit well the

training data while complying with the prior knowledge of the desired model characteristics at the same time.

  • Future work
  • Investigate various strategies to maintain the most relevant

constraint samples during the whole run.

  • Different constraints can generate violations of a very different

scale – need for some normalization.