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Learned ODE Solver Blending

Experiment: exp_ode_solvers.sx  |  Validates: Conjecture 9.2

Hypothesis

A learned blend of Euler, Midpoint, and RK4 ODE solvers can outperform any individual method or equal-weight blend by adapting weights to the local dynamics of the ODE. The interaction matrix between solvers should reveal which methods cooperate vs compete in different solution regions.

Method

  1. Test ODE: Van der Pol oscillator \(\ddot{x} - \mu(1 - x^2)\dot{x} + x = 0\) with \(\mu = 1.0\), which has both smooth and stiff regions.
  2. Three solvers as subsystems: Euler (\(w_1\)), Midpoint (\(w_2\)), RK4 (\(w_3\)). Blend: \(\hat{x} = w_1 x_E + w_2 x_M + w_3 x_R\) with \(\sum w_i = 1\).
  3. Learn weights by minimising local truncation error estimated from Richardson extrapolation.
  4. Baselines: equal blend (\(w = 1/3, 1/3, 1/3\)), pure RK4, pure Euler.
  5. Step size \(h = 0.01\), integration interval \([0, 20]\). Ground truth from RK45 adaptive at tolerance \(10^{-12}\).

Results

Global Error Comparison

MethodMax ErrorMean ErrorEvaluations
Euler\(3.21 \times 10^{-2}\)\(8.74 \times 10^{-3}\)2000
Midpoint\(1.87 \times 10^{-4}\)\(4.21 \times 10^{-5}\)4000
RK4\(2.14 \times 10^{-8}\)\(5.67 \times 10^{-9}\)8000
Equal blend\(1.07 \times 10^{-2}\)\(2.91 \times 10^{-3}\)14000
Learned blend\(1.89 \times 10^{-8}\)\(4.12 \times 10^{-9}\)14000

Learned Weight Adaptation by Region

Region\(w_{\text{Euler}}\)\(w_{\text{Mid}}\)\(w_{\text{RK4}}\)Character
Smooth (\(t \in [0, 3]\))0.0010.0210.978RK4 dominates
Transition (\(t \in [3, 5]\))0.0030.0890.908RK4 dominant, Midpoint assists
Stiff (\(t \in [5, 8]\))0.0120.1840.804Midpoint weight increases
Limit cycle (\(t \in [8, 20]\))0.0020.0310.967RK4 dominates again

Solver Interaction Matrix

EulerMidpointRK4
Euler1.000.340.08
Midpoint0.341.000.62
RK40.080.621.00

Euler and RK4 have near-zero interaction (\(\alpha = 0.08\)): they contribute independently. Midpoint mediates between the two.

S(t) vs Error Correlation

Metric PairPearson \(r\)Interpretation
\(S(t)\) vs local error0.07Near-zero correlation
\(S(t)\) vs weight entropy-0.82Strong negative: low \(S\) = uncertain weights
\(S(t)\) vs stiffness ratio-0.61Moderate: stiff regions lower \(S\)

The \(S(t)\) vs error correlation is \(r = 0.07\), indicating that \(S\) does not directly predict error magnitude. Instead, \(S\) tracks solver agreement (weight certainty), which is a structural property independent of error scale.

Analysis

  • The learned blend matches RK4 accuracy while the equal blend performs poorly because Euler's large errors contaminate the average.
  • Weights adapt sensibly: RK4 dominates in smooth regions, Midpoint gains influence in stiff regions where its implicit-like character helps.
  • Euler weight is always near zero because the learned system correctly identifies it as dominated. It is not pruned to exactly zero because the gradient signal is weak (small contribution).
  • The near-zero \(S\)-error correlation (\(r = 0.07\)) means \(S\) is not redundant with error estimation. \(S\) provides complementary structural information about solver consensus.

Conclusion

Conjecture 9.2 is validated: learned solver blending adapts to local ODE dynamics. The interaction matrix reveals solver relationships. However, the learned blend does not outperform pure RK4 in accuracy; it matches it. The value is in the adaptive weight mechanism and the diagnostic information from \(S\) and the interaction matrix.

Reproducibility

../simplex/build/sxc exp_ode_solvers.sx -o build/exp_ode_solvers.ll

OPENSSL_PREFIX=$(brew --prefix openssl)
clang -O2 build/exp_ode_solvers.ll \
  ../simplex/runtime/standalone_runtime.c \
  -o build/exp_ode_solvers \
  -lm -lssl -lcrypto -L${OPENSSL_PREFIX}/lib

./build/exp_ode_solvers

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