`, long text divided by `
`/`
` for sections. 3. Lists use `
- `, quotes use `
`, code uses `
`, important sentences use ``.
4. Preserve *all* original information, no additions/deals/changes.
5. No document-level tags (``, ``, ``, etc.).
* Input Text: A scientific news article about Physics-informed neural networks (PINNs), residual minimization, least squares estimation, and specific equations (Burgers', Cordoba-Cordoba-Fontelos).
2. Analyze the Input Text & Determine Structure:
* *Paragraph 1:* "Physics-informed neural networks (PINNs) are trained to minimize the residual of a prescribed equation, so a low training loss does not indicate whether a coefficient in that equation was specified incorrectly. This study evaluates what can be inferred about the encoded parameter from the trained network alone." -> *Main intro paragraph.*
* *Paragraph 2:* "When the residual is linear in the parameter, written as $A - νB$, the least-squares estimate $\hatν = A^{\top}B/B^{\top}B$ is obtained from a single evaluation of the frozen network, without retraining or reference data." -> *Technical detail / definition. Needs math formatting or just inline? The prompt says "semantically化的HTML片段", usually math symbols can be kept as plain text or `` if CSS is assumed, but standard HTML without classes is safer. I'll keep the math as plain text within the sentence, maybe wrap key terms like "least-squares estimate" in `` or ``.* Let's just format it nicely.
* *Paragraph 3:* "Its accuracy depends on how closely the network satisfies its own equation, quantified by a normalized residual $ρ$. Recovery error is associated with $ρ$ (loglog correlation 0.72 across 36 convection-diffusion networks), and empirical exponents of 0.63 to 1.20 across three equations decompose into one half plus a measurable angular term." -> *Accuracy details. Key terms: "normalized residual $ρ$", "loglog correlation 0.72", "empirical exponents of 0.63 to 1.20".*
* *Paragraph 4:* "For Burgers' equation, a single model test based on a block bootstrap standard error flagged 3 of 300 correctly specified networks (observed false alarm rate 1%), all 2% misspecifications at a practical budget, and two thirds of 1% misspecifications after convergence." -> *Burgers' equation results.*
* *Paragraph 5:* "For the self-similar blowup rate of the Cordoba-Cordoba-Fontelos equation, where the rate also enters the architecture, the estimator recovers about half of the misspecification, whereas $ρ$ increases approximately quadratically and distinguishes a 0.2% error." -> *Cordoba-Cordoba-Fontelos results.*
* *Paragraph 6:* "When the incorrect rate enters both the equation and the architecture, errors of up to 2% were not detected at the precision reached." -> *Limitation.*
* *Paragraph 7:* "At practical budgets, the raw loss can be lower for the misspecified model, a scaling effect that $ρ$ does not share." -> *Conclusion/Scaling effect.*
3. Drafting the HTML Structure:
* Since the prompt asks for semantic HTML with `/
` for sections, I need to identify logical sections. * Section 1: Introduction & Core Concept * Section 2: Accuracy and Residual Analysis (Linear case) * Section 3: Burgers' Equation Performance * Section 4: Cordoba-Cordoba-Fontelos Equation Performance * Section 5: Limitations and Scaling Effects * Let's refine the sectioning to make sense while strictly preserving the text. *Drafting Sections:* *
Introduction and Core Concept * `
Physics-informed neural networks (PINNs) are trained to minimize the residual of a prescribed equation, so a low training loss does not indicate whether a coefficient in that equation was specified incorrectly. This study evaluates what can be inferred about the encoded parameter from the trained network alone.
` * Linear Residual Estimation * `
When the residual is linear in the parameter, written as $A - \nu B$, the least-squares estimate $\hat{\nu} = A^{\top}B/B^{\top}B$ is obtained from a single evaluation of the frozen network, without retraining or reference data.
` * Accuracy and Residual Dependence * `
Its accuracy depends on how closely the network satisfies its own equation, quantified by a normalized residual $\rho$. Recovery error is associated with $\rho$ (loglog correlation 0.72 across 36 convection-diffusion networks), and empirical exponents of 0.63 to 1.20 across three equations decompose into one half plus a measurable angular term.
` * Performance on Burgers' Equation
* `For Burgers' equation, a single model test based on a block bootstrap standard error flagged 3 of 300 correctly specified networks (observed false alarm rate 1%), all 2% misspecifications at a practical budget, and two thirds of 1% misspecifications after convergence.
`
* Performance on Cordoba-Cordoba-Fontelos Equation
* `For the self-similar blowup rate of the Cordoba-Cordoba-Fontelos equation, where the rate also enters the architecture, the estimator recovers about half of the misspecification, whereas $\rho$ increases approximately quadratically and distinguishes a 0.2% error.
`
* Limitations and Scaling Effects * `
When the incorrect rate enters both the equation and the architecture, errors of up to 2% were not detected at the precision reached. At practical budgets, the raw loss can be lower for the misspecified model, a scaling effect that $\rho$ does not share.
` 4. Refining Semantic Elements: * Are there any lists? No bullet points in the original text. * Are there any quotes? No block quotes. * Are there any code blocks? No. * Important sentences: "low training loss does not indicate whether a coefficient... was specified incorrectly", "least-squares estimate", "recovery error is associated with ρ", "empirical exponents", "flagged 3 of 300 correctly specified networks", "observed false alarm rate 1%", "all 2% misspecifications", "estimator recovers about half", "raw loss can be lower for the misspecified model". I will wrap these in `` tags where appropriate, but must be careful not to alter the text. * Wait, the rule says "重要句子可用". It means "important sentences CAN use ", implying optional, but good practice for an "engineering" output. Let's add `` to the most critical phrases to show semantic markup. * "low training loss does not indicate whether a coefficient in that equation was specified incorrectly." -> `low training loss does not indicate whether a coefficient in that equation was specified incorrectly.` * "least-squares estimate" -> `least-squares estimate` * "empirical exponents of 0.63 to 1.20" -> `empirical exponents of 0.63 to 10.20` (Wait, typo in my thought, it's 1.20). * "observed false alarm rate 1%" -> `observed false alarm rate 1%` * "estimator recovers about half" -> `estimator recovers about half` * "a scaling effect that ρ does not share" -> `a scaling effect that ρ does not share` 5. Review against constraints: * Only output HTML fragment? Yes. * No Markdown fences? Yes. * No explanations? Yes. *
`, code uses ``, important sentences use ``. 4. Preserve *all* original information, no additions/deals/changes. 5. No document-level tags (``, ``, ``, etc.). * Input Text: A scientific news article about Physics-informed neural networks (PINNs), residual minimization, least squares estimation, and specific equations (Burgers', Cordoba-Cordoba-Fontelos). 2. Analyze the Input Text & Determine Structure: * *Paragraph 1:* "Physics-informed neural networks (PINNs) are trained to minimize the residual of a prescribed equation, so a low training loss does not indicate whether a coefficient in that equation was specified incorrectly. This study evaluates what can be inferred about the encoded parameter from the trained network alone." -> *Main intro paragraph.* * *Paragraph 2:* "When the residual is linear in the parameter, written as $A - νB$, the least-squares estimate $\hatν = A^{\top}B/B^{\top}B$ is obtained from a single evaluation of the frozen network, without retraining or reference data." -> *Technical detail / definition. Needs math formatting or just inline? The prompt says "semantically化的HTML片段", usually math symbols can be kept as plain text or `` if CSS is assumed, but standard HTML without classes is safer. I'll keep the math as plain text within the sentence, maybe wrap key terms like "least-squares estimate" in `` or ``.* Let's just format it nicely. * *Paragraph 3:* "Its accuracy depends on how closely the network satisfies its own equation, quantified by a normalized residual $ρ$. Recovery error is associated with $ρ$ (loglog correlation 0.72 across 36 convection-diffusion networks), and empirical exponents of 0.63 to 1.20 across three equations decompose into one half plus a measurable angular term." -> *Accuracy details. Key terms: "normalized residual $ρ$", "loglog correlation 0.72", "empirical exponents of 0.63 to 1.20".* * *Paragraph 4:* "For Burgers' equation, a single model test based on a block bootstrap standard error flagged 3 of 300 correctly specified networks (observed false alarm rate 1%), all 2% misspecifications at a practical budget, and two thirds of 1% misspecifications after convergence." -> *Burgers' equation results.* * *Paragraph 5:* "For the self-similar blowup rate of the Cordoba-Cordoba-Fontelos equation, where the rate also enters the architecture, the estimator recovers about half of the misspecification, whereas $ρ$ increases approximately quadratically and distinguishes a 0.2% error." -> *Cordoba-Cordoba-Fontelos results.* * *Paragraph 6:* "When the incorrect rate enters both the equation and the architecture, errors of up to 2% were not detected at the precision reached." -> *Limitation.* * *Paragraph 7:* "At practical budgets, the raw loss can be lower for the misspecified model, a scaling effect that $ρ$ does not share." -> *Conclusion/Scaling effect.* 3. Drafting the HTML Structure: * Since the prompt asks for semantic HTML with `/
` for sections, I need to identify logical sections. * Section 1: Introduction & Core Concept * Section 2: Accuracy and Residual Analysis (Linear case) * Section 3: Burgers' Equation Performance * Section 4: Cordoba-Cordoba-Fontelos Equation Performance * Section 5: Limitations and Scaling Effects * Let's refine the sectioning to make sense while strictly preserving the text. *Drafting Sections:* *
Introduction and Core Concept * `
Physics-informed neural networks (PINNs) are trained to minimize the residual of a prescribed equation, so a low training loss does not indicate whether a coefficient in that equation was specified incorrectly. This study evaluates what can be inferred about the encoded parameter from the trained network alone.
` *Linear Residual Estimation * `
When the residual is linear in the parameter, written as $A - \nu B$, the least-squares estimate $\hat{\nu} = A^{\top}B/B^{\top}B$ is obtained from a single evaluation of the frozen network, without retraining or reference data.
` *Accuracy and Residual Dependence * `
Its accuracy depends on how closely the network satisfies its own equation, quantified by a normalized residual $\rho$. Recovery error is associated with $\rho$ (loglog correlation 0.72 across 36 convection-diffusion networks), and empirical exponents of 0.63 to 1.20 across three equations decompose into one half plus a measurable angular term.
` *Performance on Burgers' Equation
* `For Burgers' equation, a single model test based on a block bootstrap standard error flagged 3 of 300 correctly specified networks (observed false alarm rate 1%), all 2% misspecifications at a practical budget, and two thirds of 1% misspecifications after convergence.
` *Performance on Cordoba-Cordoba-Fontelos Equation
* `For the self-similar blowup rate of the Cordoba-Cordoba-Fontelos equation, where the rate also enters the architecture, the estimator recovers about half of the misspecification, whereas $\rho$ increases approximately quadratically and distinguishes a 0.2% error.
` *Limitations and Scaling Effects * `
When the incorrect rate enters both the equation and the architecture, errors of up to 2% were not detected at the precision reached. At practical budgets, the raw loss can be lower for the misspecified model, a scaling effect that $\rho$ does not share.
` 4. Refining Semantic Elements: * Are there any lists? No bullet points in the original text. * Are there any quotes? No block quotes. * Are there any code blocks? No. * Important sentences: "low training loss does not indicate whether a coefficient... was specified incorrectly", "least-squares estimate", "recovery error is associated with ρ", "empirical exponents", "flagged 3 of 300 correctly specified networks", "observed false alarm rate 1%", "all 2% misspecifications", "estimator recovers about half", "raw loss can be lower for the misspecified model". I will wrap these in `` tags where appropriate, but must be careful not to alter the text. * Wait, the rule says "重要句子可用". It means "important sentences CAN use ", implying optional, but good practice for an "engineering" output. Let's add `` to the most critical phrases to show semantic markup. * "low training loss does not indicate whether a coefficient in that equation was specified incorrectly." -> `low training loss does not indicate whether a coefficient in that equation was specified incorrectly.` * "least-squares estimate" -> `least-squares estimate` * "empirical exponents of 0.63 to 1.20" -> `empirical exponents of 0.63 to 10.20` (Wait, typo in my thought, it's 1.20). * "observed false alarm rate 1%" -> `observed false alarm rate 1%` * "estimator recovers about half" -> `estimator recovers about half` * "a scaling effect that ρ does not share" -> `a scaling effect that ρ does not share` 5. Review against constraints: * Only output HTML fragment? Yes. * No Markdown fences? Yes. * No explanations? Yes. *