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    Physics-Constrained Learning for Scientific Discovery: Inference in Differential Equations and Inverse Design via Generative Models
    (University of Waterloo, 2026-08-10) Podina, Lena
    Scientific discovery increasingly relies on machine learning (ML), but many scientific problems involve sparse data, physical constraints, and large combinatorial search spaces. In these regimes, ML can suffer from generalization and robustness issues, including generating outputs that violate physics constraints known a priori. Physics-informed machine learning aims to integrate physical constraints into ML frameworks with the goal of guaranteeing physically sound outputs, especially to scientific problems. In supervised machine learning, physics-informed neural networks (PINNs) have been influential in handling problems that traditional differential equation solvers struggle with; in model discovery, symbolic regression algorithms can discover closed-form models that best describe a dataset; within generative models, generative flow networks (GFlowNets) are used to perform inverse design of drugs, antibiotics, and materials. This thesis develops physics-informed machine learning methods that integrate prior scientific knowledge into predictive, inverse, and generative models. We make three main contributions to physics-informed machine learning: a contribution in the domain of PINNs; a contribution in the domain of symbolic regression; a contribution to materials discovery via GFlowNets. In the domain of PINNs, we introduce Universal PINNs, which can be used to learn unknown components of differential equations from sparse or noisy data, and we apply them to discover the best form for the drug action of a chemotherapeutic, testing the method both on synthetic and experimental data (Chapters 2 and 3). Then, we integrate PINNs with conformal prediction, enabling PINNs to output confidence intervals with provable guarantees on both parameter fits and differential equation solutions (Chapter 4). For our contribution in the domain of symbolic regression (Chapter 5), we augment the efficiency of symbolic regression algorithms with dimensional analysis, and showcase the improvement in the performance of a well-known symbolic regression algorithm, PySR. In the domain of materials discovery (Chapter 6), we build a framework for catalyst discovery, for the application of hydrogen energy storage. In this work, we integrate ML-based relaxation, reward shaping, and action space constraints to generate stable and efficient catalysts. For two separate chemical reactions, we rediscover the best known catalysts within a constrained search space. Taken together, these contributions show that physical constraints can improve sample efficiency and reliability in scientific machine learning. Future work will integrate Universal PINNs and uncertainty quantification more tightly; GFlowNets and differential equation models could be applied together to a new scientific problem; the catalyst discovery framework could be tested experimentally and augmented with an active learning loop, in order to discover truly new materials.
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    Representing behavioural models with rich control structures in SMT-LIB
    (University of Waterloo, 2015-09-01) Day, Nancy A.; Vakili, Amirhossein
    We motivate and present a proposal for how to represent extended nite state machine behavioural models with rich hierarchical states and compositional control structures (e.g., the Statecharts family) in SMT-LIB. Our goal with such a representation is to facilitate automated deductive reasoning on such models, which can exploit the structure found in the control structures. We present a novel method that combines deep and shallow encoding techniques to describe models that have both rich control structures and rich datatypes. Our representation permits varying semantics to be chosen for the control structures recognizing the rich variety of semantics that exist for the family of extended nite state machine languages. We hope that discussion of these representation issues will facilitate model sharing for investigation of analysis techniques.
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    A Modular Notation for Monitoring Network Systems
    (University of Waterloo, 2015-07-16) Raghav, Prashant; Tefler, Richard
    Design of next generation network systems with predictable behaviour in all situations poses a significant challenge. Monitoring of events happening at different points in a distributed environment can detect the occurrence of events that indicates significant error conditions. We use a Modular Timing Diagrams (MTD) as a specification language to describe these error conditions. MTDs are a component-oriented and compositional notation. We take advantage of these features of MTDs and point out that, in many cases, a global MTD specification describing behaviours of several system components can be efficiently decomposed into a set of sub-specifications. Each of the sub-specifications describes a local monitor that is specific to the component on which the monitor is intended to run. We illustrate the compositional nature of MTDs in describing several network monitoring conditions related to network security.
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    On Fast Computation of Nested Cross-Validation
    (University of Waterloo, 2026-08-10) Cao, Shuyang
    Cross-validation is a resampling procedure that provides a point estimate of prediction error. Uncertainty in this estimate is challenging to quantify. Nested Cross-validation addresses this by providing a prediction interval. However, computing this interval can be infeasible as it requires running the cross-validation procedure and refitting the model an extraordinary number of times. Under penalized regression, we provide a fast and efficient way to compute the prediction interval with a single model fit. We give a thorough analysis of our proposed method, from the theoretical property of its core computation technique to its implementation and complexity. Our method performs best when the number of folds scales with sample size and reduce the computation by O(n), where n is the sample size. For application, we evaluate nested cross-validation on tuning parameters for signal regression models, comparing results with restricted maximum likelihood and other cross-validation based procedures.
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    Membrane development for organic solvent nanofiltration: A sustainable solution to chemical separation and purification
    (University of Waterloo, 2026-08-10) Ali, Sharafat
    Chemical separation is an integral part of the pharmaceuticals, food, petrochemicals, and fine chemicals industries for purifying and producing high-value products, extracting catalysts, and recycling organic solvents. These industries depend on conventional distillation processes for chemical separation, which are energy-intensive, costly, and generate millions of tons of CO2 annually. Membrane-based separation, particularly organic solvent nanofiltration (OSN), has emerged as an alternative low-energy separation process for separating solutes in organic media. However, current OSN membranes face challenges such as limited solvent stability, poor rejection of low molecular-weight solutes, and trade-offs between permeability and selectivity. This research was designed to address these issues through systematic design, fabrication, and modification of polybenzimidazole (PBI)-based membranes, utilizing chemical crosslinking, interfacial engineering, and supramolecular functionalization. PBI was selected as the membrane-fabrication polymer because of its exceptional thermal stability and mechanical strength, making it an ideal candidate for large-scale fabrication and deployment of OSN technology. Pristine PBI membranes could be used for separation in both polar and non-polar solvents, but they are not resistant to polar aprotic solvents, which hinder its large-scale application. Therefore, proper post-treatment modifications are required to make the membrane chemically resistant for a wide range of OSN applications. A series of PBI membranes was fabricated and crosslinked using covalent and non-covalent techniques to improve their chemical stability and selective transport behavior. In Chapter 3, interfacial oxidative polymerization of polydopamine (PDA) combined with covalent crosslinking using α,α′-dibromo-p-xylene (DBX) yielded composite membranes with excellent solvent stability. The membrane showed excellent rejection efficiency and solute fractionation while maintaining competitive permeance to organic solvents. Following the successful completion of the first study, Chapter 4 investigated the impact of chemical crosslinker chemistry on membrane performance. Two different crosslinkers with aromatic (dichloro-p-xylene (DCX)) and aliphatic (1,4-dibromobutane (DBB)) structures were used to control the interchain spacing of the PBI polymer and enhance its solvent stability. It was revealed that aromatic crosslinkers led to higher degrees of crosslinking and improved chemical resistance to strong aprotic solvents (e.g., NMP and DMAc), while aliphatic crosslinkers resulted in membranes with larger intersegmental spacing and lower chemical resistance. Subsequently, integrating with sulfocalix[4]arene (SCA4) host molecules created a framework of dual crosslinking and ionic interactions, significantly tightening the polymer network while maintaining a sharp pore size cut-off without compromising the membrane’s solvent permeance. Building upon these insights, in Chapter 5, a chemically stable PBI membrane was reported by covalent crosslinking with a novel crosslinker, triglycidyl isocyanurate (TGIC). The results revealed that the epoxide groups in TGIC underwent nucleophilic ring-opening reactions with the secondary amine groups of PBI, forming a three-dimensional polymer network with exceptional chemical resistance to both organic solvents and extreme pH conditions (pH 1 & pH 14). The membranes showed higher rejection of organic dyes with robust permeance to organic solvents and water, surpassing the available membranes. The reported membrane also demonstrated selective fractionation of mixed solutes, underscoring its potential for industrial separation processes. Overall, this thesis provides the structure-property-performance relationships of chemically modified PBI membranes involving a combination of covalent crosslinking, supramolecular assembly, and interfacial modification. This approach improves the chemical stability of PBI membranes and finely tunes selectivity and permeability at the molecular level. The findings demonstrate that PBI-based membranes can achieve the stability and performance required for industrial organic solvent separations, enabling their use for solvent recovery and fine chemical purification in large-scale industrial applications.