Sufficient statistics for parameter inference in the stochastic Van der Pol oscillator model for thermoacoustic...

Thermoacoustic instabilities pose a critical challenge in the design and operation of many combustion systems. We report a low-dimensional statistics approach for inferring the parameters of the stochastic Van der Pol oscillator from trajectory observations. From the exponential family structure of the stationary amplitude distribution, we identify sufficient statistics that capture essential parametric information. To break the parameter degeneracy inherent to stationary distribution methods, we incorporate the amplitude autocorrelation time, which provides independent information about noise intensity through relaxation dynamics. The inverse mapping from sufficient statistics to parameters can be effectively approximated by quadratic polynomial regression, achieving comparable accuracy to neural networks while requiring orders of magnitude fewer parameters and providing interpretable closed-form expressions. Comparison with previous parameter identification approaches reveals complementary strengths. In the critical regime near the Hopf bifurcation, our method enables regime classification with reasonable accuracy. Under measurement noise, a periodogram-based moment correction yields graceful degradation across all three parameter estimates, with residual high-noise error dominated by autocorrelation-time bias rather than moment bias. The methodology provides interpretable symbolic expressions suitable for thermoacoustic stability monitoring, and can be extended to other stochastic oscillator systems.