Model-based approaches for (bio)process systems often suffer from incomplete knowledge of the underlying physical, chemical, or biological laws. Universal differential equations, which embed neural networks within differential equations, have emerged as powerful tools to learn this missing physics from experimental data. However, neural networks are inherently opaque, motivating their post-processing via symbolic regression to obtain interpretable mathematical expressions. Genetic algorithm-based symbolic regression is a popular approach for this post-processing step, but provides only point estimates and cannot quantify the confidence we should place in a discovered equation. We address this limitation by applying Bayesian symbolic regression, which uses Reversible Jump Markov Chain Monte Carlo to sample from the posterior distribution over symbolic expression trees. This approach naturally quantifies uncertainty in the recovered model structure. We demonstrate the methodology on a Lotka-Volterra predator-prey system and then show how a well-designed experiment leads to lower uncertainty in a fed-batch bioreactor case study.
Figures & tables
Term
Expression
Chains
ϕ1
−(x1x2)
5
(true: −0.9x1x2 )
(x12x2)/−(x1)
1
−(x1)/(x1/(x1x2))
1
x1+((x2x1+x1)/(−(x2)/x2))
1
−<(x2x1,0.95,−0.18)
1
<(x2,0.35,−0.46)+x12/(((x1/x2)/−(x1))⋅x1)
1
Table 1: Posterior samples from Bayesian symbolic regression for the Lotka-Volterra missing physics. Here <(x,a,b)=ax+b .
Figure 1: Lotka-Volterra system. Solid lines: true state values. Thick dots: noisy measurements of states. Dashed lines: predicted states after replacing missing physics with a neural network. Dotted lines: predicted states after replacing missing physics with 10 draws from Bayesian symbolic regression.
Figure 2: Expression tree representing (2x1+1)+sin(x2) . Circular nodes contain operators ( + , < , sin ). Rectangular nodes are terminals: rounded rectangles denote features ( x1 , x2 ), and plain rectangles denote constants. The < node implements the linear transformation <(x1,2,1)=2x1+1 .
Figure 3: Bioreactor system. States and measurements.
Simplified form
μ^max
K^s
Opt. 1
Cs/(2.33Cs+11.09)
0.43
4.77
Opt. 2
non-polynomial ( exp(−Cs) terms)
—
—
Opt. 3
Cs/(2.43Cs+9.41)
0.41
3.88
Opt. 4
Cs/(2.38Cs+10.83)
0.42
4.55
Opt. 5
Cs/(2.25Cs+12.44)
0.44
5.53
True
μmaxCs/(Ks+Cs)
0.421
4.39
Table 2: Algebraically simplified equations for the bioreactor with estimated Monod parameters. The optimal experiment recovers the Monod form, while the random experiment does not.
Figure 4: Fed-batch bioreactor system. Blue solid line: missing physics, the Monod equation. Thick orange dots: predictions of the missing physics by the neural network. Green dashed lines: predicted missing physics by the top 10 trees generated by Bayesian symbolic regression.
Symbolic regression is the problem of finding an algebraic expression describing a stochastic dependence of a target variable on a set of inputs. Unlike forms of regression that fit parameters assuming a fixed model structure, symbolic regression is a search problem over the space of expressions, represented, for example, as abstract syntax trees using a library of operators. Symbolic regression is typically used in settings with limited, noisy data in the natural sciences. However, searching for a single best-fitting expression fails to capture the epistemic uncertainty about the expression, which motivates a Bayesian perspective that enables uncertainty quantification and specification of natural priors to constrain the search space. In this work, we propose ERRLESS (Entropy-Regularized Reinforcement Learning for Expression Structure Sampling), a scalable approach for sampling from the posterior distribution over expressions given data using maximum-entropy reinforcement learning. ERRLESS learns a neural policy that constructs expressions sequentially by building up their abstract syntax trees. At convergence, the policy samples expressions from the posterior. At test time, expressions can be sampled by rollouts of this policy. We demonstrate that ERRLESS achieves competitive results on the Feynman benchmark while producing short and interpretable expressions. Additionally, we demonstrate that the mean of the posterior predictive approximated by ERRLESS achieves a high coefficient of determination (R2) compared to an SMC baseline, highlighting the benefits of the Bayesian perspective in symbolic regression.
Oussama Boussif, Mohammed Mahfoud, Younesse Kaddar +6
Mila – Québec AI Institute · Université de Montréal · Independent +5
For most process systems, knowledge of the model structure is incomplete. This missing physics must then be learned from experimental data. Recently, a combination of universal differential equations and symbolic regression has become a popular tool to discover these missing physics. Universal differential equations employ neural networks to represent missing parts of the model structure, and symbolic regression aims to make these neural networks interpretable. These machine learning techniques require high-quality data to successfully recover the true model structure. To gather such informative data, a sequential experimental design technique is developed which is based on optimally discriminating between the plausible model structures suggested by symbolic regression. This technique is then applied to discovering the missing physics of a bioreactor.
Arno Strouwen, Sebastián Micluţa-Câmpeanu
JuliaHub, MA 02111-1929 USA · Biosystems Department, KULeuven, Leuven, Belgium · Faculty of Physics, University of Bucharest, Bucharest, Romania
Symbolic regression discovers explicit, interpretable equations without assuming a functional form in advance. A Bayesian approach strengthens this through probability distributions over candidate expressions, thus quantifying uncertainty in the presence of noisy and limited data. Deep Symbolic Regression (DSR) uses a neural network to generate symbolic expressions, but it is designed to identify a single best-fitting expression rather than infer a posterior distribution over models. We introduce Deep Variational Inference Symbolic Regression (DVISR), a variational Bayesian extension of DSR. DVISR replaces the original reward with the integrand of the evidence lower bound. It also extends the network architecture to output distributions over constants within expressions, enabling posterior inference over both expression trees and their associated constants. We show that DVISR can recover the true posterior in simple settings, both with and without constant tokens, and we examine how its performance changes as the size of the expression space increases. These results position DVISR as a step toward scalable Bayesian symbolic regression with uncertainty over full symbolic models.
James Butterworth, Gevik Grigorian, Alejandro DiazDelaO
Clinical Operational Research Unit University College London London, United Kingdom · Department of Computer Science University College London London, United Kingdom