id stringclasses 7
values | category stringclasses 7
values | prompt stringclasses 7
values |
|---|---|---|
CASE_01_CANONICAL_DIFFUSION | Canonical Diffusion (Dirichlet) | Consider the 1D heat diffusion equation u_t = alpha * u_xx on x in [0, L] with u(0,t)=0, u(L,t)=0, and initial condition u(x,0) = sin(pi * x / L). Given that [x] = m, [t] = s, and [u] = K, determine the physical dimensions of alpha, write the exact analytical solution u(x,t), and explicitly check if every term has unit... |
CASE_02_CANONICAL_ADVECTION | Canonical Advection | Consider the linear advection equation u_t + v * u_x = 0 on an infinite domain with velocity v in m/s, [x] = m, [t] = s, and [u] = kg/m^3. Write the general solution for initial condition u(x,0) = f(x) and verify dimensional homogeneity of the argument inside f. |
CASE_03_DIMENSIONAL_PERTURBATION_ACCEL | Dimensional Perturbation (Accelerated Field) | Consider the proposed transport equation u_t + beta * u_x = gamma * u_xx on an infinite domain, where u is temperature (K), x is in meters (m), and t is in seconds (s). Suppose an engineer specifies beta in units of m/s^2 and gamma in m^2/s. Perform a dimensional homogeneity analysis on the equation. Is this PDE physic... |
CASE_04_FRACTIONAL_ANOMALOUS_DIFFUSION | Dimensional Perturbation (Fractional Scaling) | Consider the space-fractional diffusion equation u_t = K_alpha * d^1.5 u / d|x|^1.5 with [x] = m, [t] = s, and [u] = mol/m^3. 1. What are the strictly required physical dimensions of the fractional diffusivity K_alpha? 2. If a practitioner assumes K_alpha has units of standard diffusivity (m^2/s), explain the dimension... |
CASE_05_INHOMOGENEOUS_SCALE_DRIFT | Dimensional Perturbation (Non-linear Reaction Term) | An ecological model proposes the reaction-diffusion equation c_t = D * c_xx + r * c^2, where c is population density in individuals/m^2, [x] = m, [t] = s, and D is in m^2/s. What must be the physical dimensions of the reaction parameter r? If the modeler sets r = 0.5 s^-1, evaluate whether the equation satisfies the Bu... |
CASE_06_ROBIN_REACTIVE_FLUX | Reactive Boundary (Robin Condition & Solute Loss) | A solute with concentration c(x,t) in mg/m^3 satisfies c_t = D * c_xx on [0, L] with D in m^2/s. At x = 0, c_x(0,t) = 0 (no flux). At x = L, a reactive chemical coating is modeled by D * c_x(L,t) = -k * c(L,t), with surface reaction constant k in m/s. 1. Check dimensional consistency of the boundary condition. 2. Deriv... |
CASE_07_OPEN_BOUNDARY_ADVECTIVE_LOSS | Outflow Boundary Flux | Consider advective transport c_t + v * c_x = 0 on x in [0, L] with v > 0 (m/s) and [c] = kg/m. The inlet satisfies c(0,t) = 0, while the outlet at x = L is open. Compute the time derivative of total mass M(t) = integral_0^L c(x,t) dx. Explain why the model cannot maintain constant total mass despite the absence of inte... |
CASE_01_CANONICAL_DIFFUSION | Canonical Diffusion (Dirichlet) | Consider the 1D heat diffusion equation u_t = alpha * u_xx on x in [0, L] with u(0,t)=0, u(L,t)=0, and initial condition u(x,0) = sin(pi * x / L). Given that [x] = m, [t] = s, and [u] = K, determine the physical dimensions of alpha, write the exact analytical solution u(x,t), and explicitly check if every term has unit... |
CASE_02_CANONICAL_ADVECTION | Canonical Advection | Consider the linear advection equation u_t + v * u_x = 0 on an infinite domain with velocity v in m/s, [x] = m, [t] = s, and [u] = kg/m^3. Write the general solution for initial condition u(x,0) = f(x) and verify dimensional homogeneity of the argument inside f. |
CASE_03_DIMENSIONAL_PERTURBATION_ACCEL | Dimensional Perturbation (Accelerated Field) | Consider the proposed transport equation u_t + beta * u_x = gamma * u_xx on an infinite domain, where u is temperature (K), x is in meters (m), and t is in seconds (s). Suppose an engineer specifies beta in units of m/s^2 and gamma in m^2/s. Perform a dimensional homogeneity analysis on the equation. Is this PDE physic... |
CASE_04_FRACTIONAL_ANOMALOUS_DIFFUSION | Dimensional Perturbation (Fractional Scaling) | Consider the space-fractional diffusion equation u_t = K_alpha * d^1.5 u / d|x|^1.5 with [x] = m, [t] = s, and [u] = mol/m^3. 1. What are the strictly required physical dimensions of the fractional diffusivity K_alpha? 2. If a practitioner assumes K_alpha has units of standard diffusivity (m^2/s), explain the dimension... |
CASE_05_INHOMOGENEOUS_SCALE_DRIFT | Dimensional Perturbation (Non-linear Reaction Term) | An ecological model proposes the reaction-diffusion equation c_t = D * c_xx + r * c^2, where c is population density in individuals/m^2, [x] = m, [t] = s, and D is in m^2/s. What must be the physical dimensions of the reaction parameter r? If the modeler sets r = 0.5 s^-1, evaluate whether the equation satisfies the Bu... |
CASE_06_ROBIN_REACTIVE_FLUX | Reactive Boundary (Robin Condition & Solute Loss) | A solute with concentration c(x,t) in mg/m^3 satisfies c_t = D * c_xx on [0, L] with D in m^2/s. At x = 0, c_x(0,t) = 0 (no flux). At x = L, a reactive chemical coating is modeled by D * c_x(L,t) = -k * c(L,t), with surface reaction constant k in m/s. 1. Check dimensional consistency of the boundary condition. 2. Deriv... |
CASE_07_OPEN_BOUNDARY_ADVECTIVE_LOSS | Outflow Boundary Flux | Consider advective transport c_t + v * c_x = 0 on x in [0, L] with v > 0 (m/s) and [c] = kg/m. The inlet satisfies c(0,t) = 0, while the outlet at x = L is open. Compute the time derivative of total mass M(t) = integral_0^L c(x,t) dx. Explain why the model cannot maintain constant total mass despite the absence of inte... |
CASE_01_CANONICAL_DIFFUSION | Canonical Diffusion (Dirichlet) | Consider the 1D heat diffusion equation u_t = alpha * u_xx on x in [0, L] with u(0,t)=0, u(L,t)=0, and initial condition u(x,0) = sin(pi * x / L). Given that [x] = m, [t] = s, and [u] = K, determine the physical dimensions of alpha, write the exact analytical solution u(x,t), and explicitly check if every term has unit... |
CASE_02_CANONICAL_ADVECTION | Canonical Advection | Consider the linear advection equation u_t + v * u_x = 0 on an infinite domain with velocity v in m/s, [x] = m, [t] = s, and [u] = kg/m^3. Write the general solution for initial condition u(x,0) = f(x) and verify dimensional homogeneity of the argument inside f. |
CASE_03_DIMENSIONAL_PERTURBATION_ACCEL | Dimensional Perturbation (Accelerated Field) | Consider the proposed transport equation u_t + beta * u_x = gamma * u_xx on an infinite domain, where u is temperature (K), x is in meters (m), and t is in seconds (s). Suppose an engineer specifies beta in units of m/s^2 and gamma in m^2/s. Perform a dimensional homogeneity analysis on the equation. Is this PDE physic... |
CASE_04_FRACTIONAL_ANOMALOUS_DIFFUSION | Dimensional Perturbation (Fractional Scaling) | Consider the space-fractional diffusion equation u_t = K_alpha * d^1.5 u / d|x|^1.5 with [x] = m, [t] = s, and [u] = mol/m^3. 1. What are the strictly required physical dimensions of the fractional diffusivity K_alpha? 2. If a practitioner assumes K_alpha has units of standard diffusivity (m^2/s), explain the dimension... |
CASE_05_INHOMOGENEOUS_SCALE_DRIFT | Dimensional Perturbation (Non-linear Reaction Term) | An ecological model proposes the reaction-diffusion equation c_t = D * c_xx + r * c^2, where c is population density in individuals/m^2, [x] = m, [t] = s, and D is in m^2/s. What must be the physical dimensions of the reaction parameter r? If the modeler sets r = 0.5 s^-1, evaluate whether the equation satisfies the Bu... |
CASE_06_ROBIN_REACTIVE_FLUX | Reactive Boundary (Robin Condition & Solute Loss) | A solute with concentration c(x,t) in mg/m^3 satisfies c_t = D * c_xx on [0, L] with D in m^2/s. At x = 0, c_x(0,t) = 0 (no flux). At x = L, a reactive chemical coating is modeled by D * c_x(L,t) = -k * c(L,t), with surface reaction constant k in m/s. 1. Check dimensional consistency of the boundary condition. 2. Deriv... |
CASE_07_OPEN_BOUNDARY_ADVECTIVE_LOSS | Outflow Boundary Flux | Consider advective transport c_t + v * c_x = 0 on x in [0, L] with v > 0 (m/s) and [c] = kg/m. The inlet satisfies c(0,t) = 0, while the outlet at x = L is open. Compute the time derivative of total mass M(t) = integral_0^L c(x,t) dx. Explain why the model cannot maintain constant total mass despite the absence of inte... |
Evaluating Dimensional Consistency and Boundary Invariants in Small Frontier Models
Candidate: Jefferson Alfredo Conza Fajardo
Program: Fatima Fellowship (Fall 2026 Cohort)
Evaluated Models: 'meta-llama/Llama-3.2-3B-Instruct (3.21B)' and 'Qwen/Qwen2.5-3B-Instruct (3.09B)'
Host Environment: Google Colab (NVIDIA T4 GPU)
1. Identified Blind Spot and Lived Experience Context
As an undergraduate mathematics student at Yachay Tech University researching nonstationary physical dynamics and Scientific Machine Learning, my regular workflow revolves around differential operators, boundary value problems, and conservation laws. In applied mathematical settings, symbolic expressions are invalid unless they satisfy fundamental physical invariants, specifically Buckingham Pi dimensional homogeneity and mass or energy continuity.
Standard AI benchmarks (such as GSM8K, MATH, or HumanEval) treat equations primarily as syntactic tokens, evaluating performance on canonical textbook problems with closed form solutions. However, when evaluating compact frontier models (between 0.6B and 6B parameters), a critical blind spot appears:
- The Capability Gap: Frontier compact models display strong syntactic fluency without physical grounding. While they reproduce separation of variables and Fourier expansions, they fail to track physical dimensions when parameters are modified, and they fail to enforce global conservation invariants across reactive boundaries (such as Robin or flux decay conditions).
- Practical Consequence: The model produces derivations that look plausible to non specialists but violate dimensional consistency (adding incompatible physical units) or hallucinate mass conservation in open dissipative systems.
2. Systematic Evaluation Across Compact Frontier Models
We constructed an evaluation suite of seven curated cases comparing canonical controls against dimensional perturbations and reactive boundary conditions across both Llama-3.2-3B-Instruct and Qwen2.5-3B-Instruct. The empirical logs confirm distinct manifestations of this blind spot:
Key Findings
(1) Canonical Operator Inversion and Boundary Overfitting (CASE_01): On standard heat diffusion, Qwen2.5-3B-Instruct derives incorrect dimensions for thermal diffusivity, stating [alpha] = s/m^2 instead of m^2/s. Meanwhile, Llama-3.2-3B-Instruct misapplies Dirichlet boundary evaluations to deduce a trivial null solution u(x,t) = 0, claiming smooth sinusoidal initial states are incompatible with boundary constraints.
(2) Arithmetic Hallucination of Dimensional Homogeneity (CASE_03): Under accelerated advective perturbations (CASE_03), Qwen2.5-3B-Instruct explicitly computes term dimensions as [u_t] = K/s, [beta u_x] = K/s^2, and [gamma u_xx] = K/s, yet immediately claims that all three terms have identical dimensions of K/s, demonstrating a complete breakdown between intermediate symbolic steps and evaluative reasoning.
(3) Algebraic Unit Subtraction Fallacy (CASE_02): When verifying arguments in advection solutions, Qwen2.5-3B-Instruct calculates [m] - [m] = 0 and asserts that the physical length argument x - vt is dimensionless, conflating algebraic difference with dimensionless ratios.
(4) Boundary Flux Invariant Failure (CASE_06, CASE_07): Under reactive Robin boundaries, both architectures struggle with boundary integral conservation, generating spurious self referential terms in integration by parts or failing to close global mass conservation balances.
3. Proposed Mitigation Strategy
To address this capability gap in compact models without requiring multi billion parameter scaling, we propose a three part technical approach:
(1) Symbolic Dimensional Verification in Reinforcement Learning: Integrate Computer Algebra Systems (such as SymPy or SageMath) as verifiable reward checkers during Reinforcement Learning with Verifiable Rewards. Penalize any intermediate or final derivation where terms fail dimensional consistency checks.
(2) Boundary Invariant Curated Synthetic Pretraining: Curate synthetic datasets containing domain randomized differential equations where boundary operators systematically vary across Dirichlet, Neumann, and Robin configurations, directing attention mechanisms toward boundary flux integrals.
(3) Physical Inductive Biases in Model Architecture: Incorporate explicit scale invariance preservation layers and symmetry constraints into model latent representations for scientific tokens.
Repository Artifacts
- test_cases.jsonl: Formatted evaluation prompts covering canonical controls and edge cases.
- results_llama.jsonl: Raw inference generations from meta-llama/Llama-3.2-3B-Instruct.
- results_qwen.jsonl: Raw inference generations from Qwen/Qwen2.5-3B-Instruct.
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