Symbolic regression (SR) is a machine learning approach aimed at discovering mathematical closed-form expressions that best fit a given dataset. Traditional complexity measures in SR, such as the number of terms or expression tree depth, often fail to capture the difficulty of specific analytical tasks a user might need to perform. In this paper, we introduce a new complexity measure designed to quantify the difficulty of conducting single-feature global perturbation analysis (SGPA)—a type of analysis commonly applied in fields like physics and risk scoring to understand the global impact of perturbing individual input features. We present a unified mathematical framework that formalizes and generalizes these established practices, providing a precise method to assess how challenging it is to apply SGPA to different closed-form equations. This approach enables the definition of novel complexity metrics and constraints directly tied to this practical analytical task. Additionally, we establish a reconstruction theorem, offering potential insights for developing future optimization techniques in SR.

Data-driven modeling of dynamical systems is a crucial area of machine learning. In many scenarios, a thorough understanding of the model’s behavior becomes essential for practical applications. For instance, understanding the behavior of a pharmacokinetic model, constructed as part of drug development, may allow us to both verify its biological plausibility (e.g., the drug concentration curve is non-negative and decays to zero in the long term) and to design dosing guidelines (e.g., by looking at the peak concentration and its timing). Discovery of closed-form ordinary differential equations (ODEs) can be employed to obtain such insights by finding a compact mathematical equation and then analyzing it (a two-step approach). However, its widespread use is currently hindered because the analysis process may be time-consuming, requiring substantial mathematical expertise, or even impossible if the equation is too complex. Moreover, if the found equation's behavior does not satisfy the requirements, editing it or influencing the discovery algorithms to rectify it is challenging as the link between the symbolic form of an ODE and its behavior can be elusive. This paper proposes a conceptual shift to modeling low-dimensional dynamical systems by departing from the traditional two-step modeling process. Instead of first discovering a closed-form equation and then analyzing it, our approach, direct semantic modeling, predicts the semantic representation of the dynamical system (i.e., description of its behavior) directly from data, bypassing the need for complex post-hoc analysis. This direct approach also allows the incorporation of intuitive inductive biases into the optimization algorithm and editing the model's behavior directly, ensuring that the model meets the desired specifications. Our approach not only simplifies the modeling pipeline but also enhances the transparency and flexibility of the resulting models compared to traditional closed-form ODEs.

While black-box approaches are commonly used for data-driven modeling of dynamical systems, they often obscure a system's underlying behavior and properties, limiting adoption in areas such as medicine and pharmacology. A two-step process of discovering ordinary differential equations (ODEs) and their subsequent mathematical analysis can yield insights into the system's dynamics. However, this analysis may be infeasible for complex equations, and refining the ODE to meet certain behavioral requirements can be challenging. Direct semantic modeling has recently been proposed to address these issues by predicting the system's behavior, such as the trajectory's shape, directly from data, bypassing post-hoc mathematical analysis. In this work, we extend the original instantiation, limited to one-dimensional trajectories and inputs, to accommodate multi-dimensional trajectories with additional personalization, allowing evolution to depend on auxiliary static features (e.g., patient covariates). In a series of experiments, we show how our approach enables practitioners to integrate prior knowledge, understand the dynamics, ensure desired behaviors, and revise the model when necessary.

Symbolic regression has excelled in uncovering equations from physics, chemistry, biology, and related disciplines. However, its effectiveness becomes less certain when applied to experimental data lacking inherent closed-form expressions. Empirically derived relationships, such as entire stress-strain curves, may defy concise closed-form representation, compelling us to explore more adaptive modeling approaches that balance flexibility with interpretability. In our pursuit, we turn to Generalized Additive Models (GAMs), a widely used class of models known for their versatility across various domains. Although GAMs can capture non-linear relationships between variables and targets, they cannot capture intricate feature interactions. In this work, we investigate both of these challenges and propose a novel class of models, Shape Arithmetic Expressions (SHAREs), that fuses GAM’s flexible shape functions with the complex feature interactions found in mathematical expressions. SHAREs also provide a unifying framework for both of these approaches. We also design a set of rules for constructing SHAREs that guarantee transparency of the found expressions beyond the standard constraints based on the model’s size.

Inferring unbiased treatment effects has received widespread attention in the machine learning community. In recent years, our community has proposed numerous solutions in standard settings, high-dimensional treatment settings, and even longitudinal settings. While very diverse, the solution has mostly relied on neural networks for inference and simultaneous correction of assignment bias. New approaches typically build on top of previous approaches by proposing new (or refined) architectures and learning algorithms. However, the end result—a neural-network-based inference machine—remains unchallenged. In this paper, we introduce a different type of solution in the longitudinal setting: a closed-form ordinary differential equation (ODE). While we still rely on continuous optimization to learn an ODE, the resulting inference machine is no longer a neural network. Doing so yields several advantages such as interpretability, irregular sampling, and a different set of identification assumptions. Above all, we consider the introduction of a completely new type of solution to be our most important contribution as it may spark entirely new innovations in treatment effects in general. We facilitate this by formulating our contribution as a framework that can transform any ODE discovery method into a treatment effects method.

Transparent machine learning (ML) models are essential for ensuring interpretability and trustworthiness in decision-making systems, particularly in high-stakes domains such as healthcare, finance, and criminal justice. While transparent machine learning models have been proposed for classification and regression, time series forecasting presents some unique challenges for ensuring transparency. In particular, currently used bottom-up approaches that focus on the values of the time series at specific time points (usually regularly spaced) do not provide a holistic understanding of the entire time series. This limits the applicability of ML in many critical areas. To open up these domains for ML, we propose a top-down framework of bi-level transparency, which involves understanding the higher-level trends and the lower-level properties of the predicted time series. Applying this framework, we develop TIMEVIEW, a transparent ML model for time series forecasting based on static features, complemented with an interactive visualization tool. Through a series of experiments, we demonstrate the efficacy and interpretability of our approach, paving the way for more transparent and reliable applications of ML in various domains.

Real-world machine learning systems often encounter model performance degradation due to distributional shifts in the underlying data generating process (DGP). Existing approaches to addressing shifts, such as concept drift adaptation, are limited by their *reason-agnostic* nature. By choosing from a pre-defined set of actions, such methods implicitly assume that the causes of model degradation are irrelevant to what actions should be taken, limiting their ability to select appropriate adaptations. In this paper, we propose an alternative paradigm to overcome these limitations, called *self-healing machine learning* (SHML). Contrary to previous approaches, SHML autonomously diagnoses the reason for degradation and proposes diagnosis-based corrective actions. We formalize SHML as an optimization problem over a space of adaptation actions to minimize the expected risk under the shifted DGP. We introduce a theoretical framework for self-healing systems and build an agentic self-healing solution *$\mathcal{H}$-LLM* which uses large language models to perform self-diagnosis by reasoning about the structure underlying the DGP, and self-adaptation by proposing and evaluating corrective actions. Empirically, we analyze different components of *$\mathcal{H}$-LLM* to understand *why* and *when* it works, demonstrating the potential of self-healing ML.

Closed-form differential equations, including partial differential equations and higher-order ordinary differential equations, are one of the most important tools used by scientists to model and better understand natural phenomena. Discovering these equations directly from data is challenging because it requires modeling relationships between various derivatives that are not observed in the data (equation-data mismatch) and it involves searching across a huge space of possible equations. Current approaches make strong assumptions about the form of the equation and thus fail to discover many well-known phenomena. Moreover, many of them resolve the equation-data mismatch by estimating the derivatives, which makes them inadequate for noisy and infrequent observations. To this end, we propose D-CIPHER, which is robust to measurement artifacts and can uncover a new and very general class of differential equations. We further design a novel optimization procedure, CoLLie, to help D-CIPHER search through this class efficiently. Finally, we demonstrate empirically that it can discover many well-known equations that are beyond the capabilities of current methods.